I think the canonical resource to this topic is:
Achterberg, Tobias. Constraint integer programming. Diss. 2007.
As this basiclly describes the theory behind the open-source MILP solver:
- SCIP (nowadays more than MILP)
it's not surprising, that there are some rather complex applications of constraint-propagation, even related to scheduling, where things get complex very fast because CP community introduced lots of very powerful propagation-mechanisms.
See for example:
Heinz, Stefan, Wen-Yang Ku, and J. Christopher Beck. "Recent improvements using constraint integer programming for resource allocation and scheduling." Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems: 10th International Conference, CPAIOR 2013, Yorktown Heights, NY, USA, May 18-22, 2013. Proceedings 10. Springer Berlin Heidelberg, 2013.
(it's more than "just" propagation however: e.g. conflict-analysis)
So yes... You can add domain-propagation to MILP (if the solver is extensible enough).
One direction, related to scheduling and probably to your example domain, which i never saw in code, but which seems like a good idea to me (depending on the problem to solve of course) would be introducing the domain-propagation of Simple Temporal Networks, for which bounded incremental algorithms are available (as propagation from scratch is equivalent to the all-pairs shortest-paths problem: rather expensive).
I'm pretty sure, "IBM ILOG CP optimizer for scheduling" does exactly this and more, which is indicated in:
Laborie, Philippe, et al. "IBM ILOG CP optimizer for scheduling: 20+ years of scheduling with constraints at IBM/ILOG." Constraints 23 (2018): 210-250.
Keep in mind, that this software seems to be a hybrid of CP and MILP (although the how / tightness of this integration is hard to estimate as it's commercial software).
Remarks about heuristics
If heuristics are compatible with domain-propagation surely depends on details.
In CP and CIP (Constraint Integer-Programming) community, i'm pretty sure the following (in absense of a better-suited resource) holds:
Lagerkvist, Mikael Zayenz, and Magnus Rattfeldt. "Half-checking propagators." arXiv preprint arXiv:2007.05423 (2020).
"Propagators are required to be correct; they must
never remove a value from a variable that may still be a solution to the constraint.
This means that propagation is not actually concerned with finding a solution
but about proving that no solution exists for a certain variable-value pair, which
is a subjectively harder problem."
So everything happening during domain-propagation basically assume a closed-world model (current domain state, current constraint-state) and one has to be careful about the implications.
If you shrink domains by heuristics and continue, the solver surely can exploit follow-up domain-propagation much more heavily and speed up reasoning / local optimization. But you lost global-optimality of course (as you assumed something without knowing if you are right). This sounds like something focused on the generation of "fast good solutions". In some sense, one might see this as primal-heuristic (which afaik in SCIP can be interleaved with constraint-propagation).
There is probably a lot possible in this direction although semantics might deviate a lot!
If you are able to do state-restoration / backtracking, you can embed this in some outer tree such that you don't lose global-optimality.
Something similar and yet very different
There might be a connection to the concepts of: probing (in MP / CIP community) or singleton arc-consistency / shaving (CP community) although i fail to "build a bridge".
Shaving is actually something very popular in scheduling applications, but shaving (as i understood it) is solely used to refute parts of the search-space which is pretty much the oppositve of our above "fast good solution" approach.