The standard Bayesian parameter estimation deals with the problem of estimating $P(\theta|x_{1:k})$, where $\theta$ is a constant but unknown parameter and $x_{1:k}$ are observations/data. I know there exists some convergence results showing the point estimate, say MLE $\hat{\theta}$, converges to the true parameter value $\theta$.
My question is that, if $\theta$ is time variant and we know its deterministic dynamics, e.g. $\dot{\theta}=f(\theta)$, can we obtain some similar result that the error between the MLE $\hat{\theta}_t$ and the true value $\theta_t$, i.e. $e_t=\hat{\theta}_t-\theta_t$, converges to $0$ asymptotically?