Also by analogy with continuous-time Markov processes, it might be easier to forget about individual trajectories and instead think about the "flow" of probability density in phase space which can probably be described by a partial differential equation without needing to define your own Riemann sums, path integrals and such. Or maybe I'm missing something here and you really need customized machinery?

It might work. I haven't thought about it yet.

Also Cosma Shalizi is an expert on both causal models and continuous-time stochastic processes, so maybe you could ask him or look at his work if you haven't seen it already.

I know of him and have read some of his stuff, but the work isn't in a sufficiently stable state to bother an academic with it yet. I need more evidence that this is the fruitful path. I expect it would be difficult to convince him of the value of such an effort, since there's no evidence yet that it's even different from what is being done already.