If continuous-time Markov models contain a parameter q that denotes the instantaneous rate of change and a global/strict clock model is the overall rate of evolution, how are these two parameters related? That is, for a given substitution model (e.g., JC69, F81, GTR), is there a deterministic relationship between the rate parameter(s) of these substitution models and the rate of the clock model? For instance, should the individual instantaneous rates of change (e.g., A -> C, T -> C, etc.) sum up to the global clock rate? If this is not the case, how should one understand the relationship between the rate of a continuous-time Markov model and the overall evolutionary rate?
Essentially you are comparing a "local clock" with a global clock and whether the instance rate relates to the global it depends on the model and in MCMC clocks in most cases the answer is no. A global clock is a very specific and very limiting model (below). Everything below is a comparison between taxa not between alignment positions.
"Local clock" in the context of MCMC clocks is in inverted commas because they are using population models. I will use the nomenclature here for convenience to describe rate heterogeneity between branches, not within a branch (last example).
A "local clock" has mutation rate heterogeniety across the tree whilst a global clock is completely homogenious. You can't say how is a "local clock" related to global clock because there are numerous different models.
Global clocks never hold for biological data because there are always some taxa evolving at a faster rate than others. This is particularly true in infectious disease.
One historic model used in traditional local clocks (no inverted commas) was autocorrelation, where a local clock would relate to a global clock on a "clade by clade" basis. That is clades are correlated by their mutation rate, so some clades evolve faster and other clades slower. Statastically an autocorrelated clock always outperforms a global clock under a likelihood ratio test.
I remember autocorrelation clearly because I implemented a variant of this approach, so I needed to understand how they did it, rather than click a few buttons on a GUI, which is what happens in Beast.
Beast has at least 3 population models and off-hand I cannot remember the population scenarios. I do remember the population scenarios made sense and secondly they can give 10-fold differences in rate in the data set I looked at, You need to read on the background theory for each model and assess what is more biologically appropriate for a given context. One model (from dim memories) reminded me of a classic population growth equation, i.e. an exponential. Population growth and collapse in phylodynamics would be informative of the rate, but again you need to go to the underlying theory.
The molecular clock theory that the rate of evolution towards the tips of trees is much faster than at the deeper branches is the sort of model that could be implemented in a MCMC clock. In the latter case the relationship between a global clock and the rate heterogeneity clock (which is a better description in that model than "local clock") is quite different from global vs. "local clock".
The fundamental issue with clocks is exemplified in the 10-fold difference issue. The lesson that is important to me is very easy to get a "wrong answer". The lesson important to you is if two MCMC models can't agree the same rate how can both obtain the same answer as a global clock? I am personally a fan of maximum heterogeneity, but in this day and age they will ask you to perform a Beast model to support it.