Neighbourhood in local search metaheuristic
I think the property of being able to reach all the successor states from a current state is crucial and needed in order for the problem to be well-defined. If this was not the case, you might miss out on good solutions. Moreover, you might never reach a goal state and hence run forever.

The classic AI book by Russell &amp; Norvig, 3rd edition, page 67 gives a formal definition for a problem. One of the components is a successor function which returns all the successor states for a current state. Together the formal components create a search space, that is the set of all states reachable from the initial state. If one uses a different successor function, namely such that it leaves out some successors, the resulting search space is a different one: goal state(s) might be missing and optimal solutions might very well differ. Perhaps the reason you are unable find it from the literature is that the requirement is obvious.