Modeling count data with time-dependent rate

For processes of discrete events occurring in continuous time with time-independent rate, we can use count models like Poisson or Negative Binomial. For discrete events that can occur once per sample in continuous time, with a time-dependent rate, we have survival models like Cox Proportional Hazards.

What can we use for discrete event data in continuous time where there is an explicit time-dependence that we want to learn?

I understand that sometimes people use sequential models where each node is the predicted time to the next event. But this doesn’t have the robustness of something like a rate function in survival analysis, where you can express a probability of event over any given time window. Ideally I would like to express a probability distribution over counts for any custom time window, with the model.