Suppose we have a Metropolis-Hastings sampler for a target distribution $f$, and we use a proposal density $Q_t$, that may depend on time $t$. By construction, $f$ is still an invariant density of the chain, at every time step (*). However, the chain is not a Markov chain anymore, at least not a time homogeneous one. I'm pretty sure there exists some theory that ensures that the process still converges to $f$, but can not find it in standard references. Where can such results be found?
* We assume that basic non degeneracy conditions hold, for instance both $f$ and all $Q_t$ are non-negative on the whole state space.