I think it should be possible to set a preferred assistant manager for managers in the database, resulting in said manager trying to bring in their preferred assistant whenever possible. I don't think managers bringing their assistant with them to new clubs or countries that they start managing happens often enough - think the only example of it happening naturally is Jurgen Klopp bringing Zeljko Buvac with him to Man City on my most recent save.
Other than that, I haven't really seen it take place at all. For example, current Port Vale manager Neil Aspin has had Lee Nogan as his assistant manager since his time managing Halifax, having brought him to both Gateshead and Vale as well, yet in game Aspin just doesn't seem to bring Nogan with him whenever he gets a new job despite Nogan being in his favoured personnel. Another example is Billy Heath - he and Mark Carroll having been working together since 2002 at Bridlington, Frickley, Hall Road Rangers, North Ferriby, Halifax and Alfreton, but Heath does not attempt to bring Carroll with him whenever he gets a new job despite Carroll being in his favoured personnel.