  We consider first-order methods with constant step size for minimizing
locally Lipschitz coercive functions that are tame in an o-minimal structure on
the real field. We prove that if the method is approximated by subgradient
trajectories, then the iterates eventually remain in a neighborhood of a
connected component of the set of critical points. Under suitable
method-dependent regularity assumptions, this result applies to the subgradient
method with momentum, the stochastic subgradient method with random reshuffling
and momentum, and the random-permutations cyclic coordinate descent method.
