I guess the big thing you gotta decide is which parameters you want to depend on time.
With respect to this zero inflated beta thread: Zero one inflated beta regression in STAN, I guess you could make either \alpha (the mixing distribution) or \gamma (the bernoulli probability) depend on time.
At that point you can pick a time series model for that parameter and rock and roll maybe? If the underlying time series is written in terms of normals, then maybe you gotta add a link function in there like:
\gamma_t = \text{inv_logit}(\mu_t)
if \mu_t is your time series model.