Hey Titian.

If you want to make your parameter change with respect to other parameters you can do that if you wish but you will be looking at a completely different distribution.

The classic way that this is talked about in textbook, research papers, and among practitioners is by referring to it as a stochastic process. It can be in either fixed intervals (discrete time) or instantaneously changing (continuous time).

Your problem is looking at a collection of random variables in your dimensions that change over time and space.

If you want to look at say calculus or means and other moments across the means of variation (time and space) then you need to look at stochastic calculus and understand the basic concepts and techniques in that field.

The introduced method deals with Brownian Motion and Wiener Processes but you can find examples where they look at Poisson models. Your model will be a complex one and you may have to derive a lot of the stuff yourself, but the ideas in these resources should help you understand the terminology and also to get started in getting a solution to your problem.