NOTE:  My apologies, as this ended up being substantially longer than I anticipated.  It's hopefully an interesting read though, and I think it would be worthwhile to at least Ctrl+F and see where BYU shows up, even if you don't want to read the whole thing.

Bold represents the falling apart history of the Big East. 
Italics represent the situation as it has unfolded in the Big 12.
We'll be jumping back and forth to highlight the parallels, but if you prefer, you can read each story separately by skipping the other conference's sections.

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In 2003, the Big East lost 3 important members (Miami, Boston College and Virginia Tech) to other AQ conferences.

In 2011-2012 the Big 12 lost 4 important members(Nebraska, Colorado, Mizzou, Texas A&M) to other AQ conferences.

Back in 2003, the Big East replaced these with 3 schools (Cincinnati, South Florida, and Louisville) from weaker conferences, and had a period of surprising stability despite the earlier defections. During this time, they even considered expansion, but opted against taking the strongest candidate (UCF) due to political issues (specifically USF continually blocking UCF's admittance).  7 years after losing key members, in 2010, the Big East ultimately settled for a small-than-anticipated expansion (TCU only). 

Back to 2012, the Big 12 replaced their lost 4 schools with two (TCU and West Virginia) from a smaller/weaker conference, and had a period of surprising stability despite the earlier defections.  During this time, they even considered expansion, but opted against taking the strongest candidate (BYU) due to political issues (specifically, concern about BYU's affiliation with the LDS church and stance on LGBTQ+ issues).  5 years after losing key members, in 2016, the Big 12 ultimately opted against an anticipated expansion.

Following the addition of TCU, things looked to be trending up for the Big East ... until the next round of TV rights renegotiations. After failing to get an offer on par with the PAC12, the Big East voted against a deal with ESPN. The resulting instability led two important, founding members (Syracuse and Pitt) of the conference to defect.

Similarly, things looked to be trending up for the Big 12 (with the PAC12 falling into role as "weakest" conference) ... until the next round of TV rights negotations.  After failing to convince ESPN and FOX to revisit TV negotiations early, the resulting instability led two important, founding members (Texas and Oklahoma) of the conference to defect.

Smelling blood in the water, other conferences poached two additional members of the Big East - with TCU and West Virginia leaving the conference (TCU without ever having played a year in it)

For the Big 12, we've reached the point where actions are TBD.  There's talk that other schools may be poached.  If so, they'll continue to follow the path of the Big East.  Whether that's Kansas to the Big 10, West Virginia to the ACC, or any other team/conference pairing, any further defections would be disastrous.  That said, it's a certainty that the current Leftover 8 are actively looking for landing spots.

In panic and fighting for survival, the Big East sent out a mass of invites, including to those they'd earlier turned down for political reasons.  UCF, Memphis, Houston, SMU were invited as full members and BYU, Boise, San Diego St, Air Force and Navy were invited as football-only members.  Adding these 9 was seen as a way to salvage the conference despite their losses, and would leave the conference with 16 teams.

Looking over the internet and CB, you see a lot of suggestions for the Big 12 to expand in the same way, up to 14 or 16.  Even a lot of the teams are the same: UCF, Cincy, BYU, Houston, Boise are the most commonly suggested names for the new Big 12 - only Cincy is new.  Additional teams such as Memphis, San Diego St, SMU and Air Force were considered both then and now.

In the Big East's fight for survival, 7 of the 9 teams invited accepted invitations.  BYU was unwilling to work with the conference financially and relinquish TV rights to the conference (perhaps one early reason for BYU's reputation as difficult to work with), so they and Air Force were left out.  It appeared that the Big East would survive, albeit drastically weakened, now as a 14 team football conference, with the potential to go to 16.  The bleeding had been stemmed.

Again, no way to discuss this for the Big 12 as they haven't yet sent out invitations, let along had them accepted.  It is worth noting that this step wasn't enough to save the Big East though, so may also not be enough to save the Big 12.

Armed with reinforcements who were happy to finally be in an AQ conference and have access to BCS bowls, the Big East had survived 2010.  They would play a few seasons as an 8 team conference, but would go up to 14 with their reinforcements.  In 2012, the Big East even added Temple and paved the way to add Villanova, meaning that, as soon as the next season, they could be a 16 (football) team AQ conference. 

And then 2012 came.  With early rumblings that the TV deal might not be (anywhere close to) what members would want, schools that had committed to join in 2013 started having second thoughts.  Schools in the conference again developed wandering eyes.  Rutgers left to join the Big 10, and Louisville to the ACC.  In the span of a decade the Big East had now lost Miami, Boston College, Virginia Tech, Syracuse, Pitt, TCU, West Virginia, Rutgers and Louisville.  Only Temple remained from the 2002 football membership (and only after having been kicked out and then re-invited)

The Big 12 has, in the last decade, lost 6 of 12 in Colorado, Nebraska, Mizzou, Texas A&M, Texas, Oklahoma.  Again, if they can keep everyone else, they should be OK, but losing any additional is probably a death blow to the conference.  That said, every member of the conference is (and should be) looking elsewhere in support of their own interests.  Prisoner's dilemma situation.

Even with those astounding 9 losses, the Big East was likely to pull through as a conference.  They simply reached deeper into the bag of G5s and added East Carolina and Tulane, and, in 2013, had an anticipated future football membership of Cincinnati, UConn, USF, UCF, Memphis, Houston, SMU, Boise, San Diego St, Navy, East Carolina, Tulane, and Temple.  Not great, but even while losing basketball schools and rebranding as the AAC, they'd retained their AQ status.  At this point, turnover didn't matter; so long as they kept replacing teams they'd keep their status.

The kicker came, not from any realignment then, but from negotiations on a new college football postseason.  With increased BCS access for G5 teams (and lower than anticipated AAC contracts), Boise St and San Diego St no longer felt it as important to join an AQ conference.  Boise St was seen as the linchpin to AAC survival, since if they stayed, San Diego St was likely to stay as well. The AAC was also recruiting BYU, Air Force, Fresno St and UNLV to join, but BYU and Air Force had indicated that they would stay and Fresno and UNLV would only join if Boise stayed.  This made Boise very important as their membership would swing the membership of San Diego State, UNLV and Fresno as well.

Eventually Boise and San Diego St, along with BYU, entered into negotiations to rejoin the MWC and ultimately left the AAC without ever playing a game.  Boise St was able to swing some concessions from the MWC that benefit them to this day, but BYU (unlike the other two) was unable to reach a deal with the MWC.

This left the AAC largely as we know it today (Tulsa/UConn have since been swapped in/out) and in the next iteration of college postseason play (CFP), the much-weakened AAC was left out and designated as a G5, rather than a P5.

This has interesting parallels for the Big 12 as negotiations are once again underway to modify the college football postseason.  This may make it more difficult for the Big 12 to recruit far-flung programs who see no benefit in being in a conference without considerably better postseason access or TV Contracts.  The advantage that the Big 12 does have over the Big East/AAC in this case is twofold - 1) there shouldn't be any confusion with name changes (due in the Big East only to the basketball/football school dichotomy which has been laregly ignored in this discussion) and 2) there may not be any concrete P5/G5 designation or discussion of auto-bids in the new system.  This is TBD though as NY6 bowls may become playoff locations, but many others will likely retain conference affiliations.

CONCLUSION: If you've made it this far, thank you - this has been far longer than I anticipated, but it's been an interesting ride for me.
Looking through the demise of the Big East as a football power, it stands out to me that aggressively adding new teams didn't really do much for them.  Yes, they "survived", albeit as the AAC that we know today and with almost none of the history or prestige of the Big East.  While the Big 12 should certainly try to bring in new programs, the time for that was ideally 5+ years ago.  Stuffing the conference full of newcomers won't do much.  The key will be retaining whatever current members they can to create some feeling of continuity.  In some respects this is easier as other conferences have less motivation to poach the remaining teams, but if your best defense is that nobody wants your teams, that's a pretty poor spot to be in.  

As pertains to BYU, it's interesting to consider a few aspects:

1) If BYU had joined the Big East, as invited, in 2011 could that conference have survived and similarly retained Boise St and San Diego St?  Would that stability have been enough to keep the TV payouts a bit higher and prevented the loss of Rutgers and Louisville?  It's a possibility and may have been enough to keep the Big East/AAC recognized as an actual P6 with the postseason advantages thereof.

2)  BYU really has had a bad go of it in negotiations.  I'm not sure if this is because BYU overvalues their program or if it's just BYU being unwilling to be lowballed, but BYU has had quite a few negotiations fall through.  Below is a short list of failed negotations for BYU to join a conference

2a) The Big 8 (precursor to the Big 12, but without the Texas schools) had seriously considered adding BYU in 1993, either standalone or in conjunction with the Texas schools from the SWC.  This obviously didn't happen, with substantial reporting that it ultimately came down to BYU vs Baylor and political pressures (Texas Governor) pushed Baylor through. There may (or may not) be more as it would have been possible to take both, but ultimately we don't know all the details.

2b) The Big 12 looked at expansion in 2010/2011.  BYU was considered, but ultimately TCU was selected.  There have been rumblings about BYU being difficult to work with, but the reason for BYU not receiving an invite is a hotly contested mystery

2c) The Big East attempted to add BYU in 2011.  7 other schools accepted the invitation, but BYU refused to give up rights to their home football games.

2d) In 2012, BYU entered negotiations (along with Boise St and San Diego St) to rejoin the Mountain West Conference.  The other two were able to rejoin and Boise even received several concessions (their games were sold in a separate package from the rest of the conference and a larger percentage of post-season payouts went to the teams earning them).  This makes for an interesting comparison between BYU and Boise St in the MWC with their advantges. 

These have led Boise St to have 69.4% of their games nationally televised over the last 4 years, very similar to BYU's 70.6%.  In 2019, BYU's athletics brought in $72.6M dollars, which was substantially more than Boise's $51.9M, although the football revenues may have actually favored Boise, with $23.2M of BYU's revenue reported as coming from football (link) and Boise football bringing in $24.8M (link).  I'm not sure how accurate that reported BYU football revenue is (though I'm sure BYU's other sports bring in more than Boise's so those should explain at least part of the $20M difference), but it makes for an interesting thought about paths not taken.

2e) In 2016, BYU was invited to apply to the Big 12, but ultimately did not receive an invite for expansion.  Again, the exact causes are a hotly debated mystery and are likely some combination of financial realities, on-field performance, geography, reputation as difficult, and potentially social pressures.