When you move your equities to a DAF, you have donated something to a charity without conditions. So, any equities you donate are liquidated in the process. It shows up in your DAF as cash initially (or maybe when you make the donation, you direct the investment - I don't remember). So, you can't self direct with unlimited choice, but you can direct some limited choices like:

Money Market
Conservative
Aggressive
etc. (My purpose is to donate shortly, so I didn't really pay attention - will always select the money market. If I wanted to invest it, I would keep it in my account).

This is Schwabs method - it may be different at others, but hard to believe you could have a self directed DAF...but anything is possible. Bottom line is in this one, you could invest it, and it could increase or decrease.