A Markov process $X_t$ on $E$ is a Feller-Dynkin (or sometimes just Feller) process if its semigroup is a strongly continuous, sub-Markov semigroup $\{P_t:t\geq 0\}$ of linear operators on $C_0(E)$ (my definition comes from the text of Rogers and Williams or see Wikipedia).

I am wondering what is known about when a time change of a Feller-Dynkin process is still Feller-Dynkin. I believe that I have read that this is not true in general (and can be a tricky thing to determine). I would be curious to see an example where this fails and also to know if the Feller-Dynkin property is preserved if the time change is nice enough.

For what it's worth, the time changes I am interested in are actually $C^{\infty}$. I do not need the Feller property to be preserved, but thinking about it made me realize I do not have a very good feeling for these sort of things.