Is there any known work on generating random uniformly distributed DAGs given a set of path existence/absence constraints?
There is some work concerning random orders. Good and slightly outdated survey is $[1]$. In this survey you can find description for the basic models of random orders: uniform model, model of random graphs orders and model of random $k$-dimensional orders. It should be noted that these models are derived for arbitrary random orders and maybe you won't find there results on generation of random orders which are extensions of the given partial order.


Brightwell G. Models of Random Partial Orders. Surveys in combinatorics, 1993.