I am currently trying to make some small parts of the minimal model program work for some very explicit varities in positive characteristic. I have such a variety $X_1$ and I know that there is a resolution of the singularities $f_1: V_1 \to X_1$. Moreover, I have another variety $X_2$ which I don't know much about at all in the same birational equivalence class. There is also a resolution $f_2: V_2 \to X_2$ for that variety. What I want to have is a single smooth variety $V$ that resolves the singularities of $X_1$ and $X_2$.

Is anything known about such problems? Any conditions under which something like this could exist would also help me alot. I know that the varieties $V_1$ and $X_1$ have some nice properties concerning Frobenius splitting. So at least for those, a lot of cohomology vanishing results are true.

Moreover, I try to find out what people expect for resolution of singularities in characteristic $p$. The result in characteristic $0$ for example extends to the problem above. If I cannot resolve this problem, I would at least like to prove things under some conjecture that might be true.