Many works combining MILP and CP are what's sometimes called logic-based benders decomposition and there is even a book on the topic:
Hooker, John. "Logic-Based Benders Decomposition: Theory and Applications." (2023).
Besides Hooker who probably coined the term, most of the work i read (scheduling/routing) was from Christopher Beck
For a good background (when shift from master to sub-problem solver: basically with primal feasible solution or only when reached an optimal one), the following might also be a worthwile read:
Beck, J. Christopher. "Checking-up on branch-and-check." Principles and Practice of Constraint Programming–CP 2010: 16th International Conference, CP 2010, St. Andrews, Scotland, September 6-10, 2010. Proceedings 16. Springer Berlin Heidelberg, 2010.
A high-quality implementation (linked to papers and a dissertation) which is a bit more far-reaching (and therefore introducing some complexity) would be:
This and some others howewer are using CP/SAT-Hybrids as their sub-problem solver component aka "CP" (or-tools cp-sat is also a CP/SAT/MILP-hybrid).
Among other differences to pure CP, these hybrids are very good at efficiently inferring conflict-clauses (because of SAT-style conflict-analysis) which is a very good thing in LBBD because the cuts will be much more accurate.
Imagine a vehicle-routing problem where a MILP does assign routes to vehicles and the sub-problem solver not only recognized that this tour of 11 stops isn't feasible (e.g. duration or time-windows) but efficiently infer that out of those 11 stops, those 3 alone are a source of infeasibility.
Compare with some or-tools cp-sat API:
// A list of literals. The model will be solved assuming all these literals
// are true. Compared to just fixing the domain of these literals, using this
// mechanism is slower but allows in case the model is INFEASIBLE to get a
// potentially small subset of them that can be used to explain the
// Think (IIS), except when you are only concerned by the provided
// assumptions. This is powerful as it allows to group a set of logically
// related constraint under only one enforcement literal which can potentially
// give you a good and interpretable explanation for infeasiblity.
// Such infeasibility explanation will be available in the
// sufficient_assumptions_for_infeasibility response field.
repeated int32 assumptions = 7;
The paper linked to the implementation even claims, that this general-purpose conflict-analysis discovers the same cuts as specialized TSP/VRP ones developed specifically for these tasks.