In parallel tempering I have replicas of the Markov chain I'm studying evolving at different temperatures, and intermittently I swap the replicas on a nearest-neighbouring temperatures basis. Between swaps I evolve the replicas independently for $M$ Monte Carlo steps. Is it possible to set $M$ too small and stop the algorithm from reaching equilibrium? Or just cause it to perform worse than independent Markov chains?
The parallel tempering principle is valid for $M=1$, so one cannot switch " too often". When the swaps are balanced (as in Neal, 1999), the product of the powered targets is the stationary distribution of the multi-chain and the target of interest the stationary of the corresponding sub-chain. On the opposite if $M$ is large, swapping occurs very little and hence this sub-chain moves very little.
Remember that parallel tempering is used to improve the mixing of the lowest temperature chain. The other chains are just auxiliary chains. But in any case when swapping occurs it is because the value of a neighbouring chain is also acceptable for the current chain. Hence there is no reason to infer a reduction in efficiency (except for the added time in simulating all these chains).