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When performing a local search, for example, a 2-node exchange in a metaheuristic (like GRASP), what is the usual stopping criteria for that procedure? Some bibliography says that the local search stops when no further improvement can be found in the neighboorhoud of the current solution. My question is: whether it is correct to stop the local search as immediately as a first better solution is found?

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The stopping criterion depends on the specific algorithm used; some common ones include after a fixed number of iterations, or after the total improvement over the past $t$ iterations is at most $u$ (for some parameters $t,u$).

No, it's not correct to stop after you make the first improvement. Usually you will need many iterations of improvements.

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