I was wondering what goes into the consideration of choosing the frequency to attempt swaps for replica exchange? D. Sindhikara, 2008 proposes the faster the better, but this goes against the view that you should allow the systems to relax to their new conditions.
The sampling enhancement in replica exchange (REMD) comes completely from transitions between your different distributions, so the more transitions you attempt, the faster you will achieve equilibrium between all your different distributions. However, MD doesn't start generating locally decorrelated samples before ~1-10 ps of sampling (the true number will depend on the nature of the system and the topology of the local minimum you are exploring), so below this timescale you are very unlikely to observe any transitions of interest, even in your most "mobile" replica. Your samples are never completely decorrelated in the global sense, however, but this is something we accept and is the reason we use REMD in the first place.
Therefore, any swap attempts after less time than your decorrelation time are still "correct" in the sense that they will converge to the right distributions but they are very unlikely to provide any increase in sampling quality (remember that the only transitions of interest are the ones that won't occur in the replica of interest in that timescale). And as was already stated, swaps need energy evaluations, which need CPU time / computational overhead and if you attempt swaps below your decorrelation time, while not technically wrong, you are quite literally wasting your time. If that computational overhead was not needed, there wouldn't be any reason not to run swaps at each timestep - there would be nothing to lose by doing that.
So I would say anything between 1-10 ps is fair game in the sense that in order to obtain the optimal parameter, you need to have knowledge which you can only obtain by running enhanced sampling in the first place! This agrees with the idea of "relaxation", which actually just means that you obtain a typical sample around your local minimum which has no local memory of the previous sample. Again, global memory is still there, so "relaxation" (and "local") is fundamentally a vague and timescale-dependent term - the space you sample locally within 1 ps is much more narrow than the space you sample within 1 ms.
As with many such hyperparameters, there is no hard and fast rule and you will probably have to do some testing to see what is best. The swap attempts may come with substantial overhead, which might be something to consider as well.
Disclaimer: my experience with replica exchange comes from quantum Monte Carlo, not MD.