Basically, a function is Lipschitz continuous if it has a bounded slope. This notion can be extended to set-valued maps in different ways. We will mainly focus on one of them: the so-called Aubin (or Lipschitz-like) property. We will employ this property to analyze the iterates generated by an iterative method known as the proximal point algorithm. Specifically, we consider a generalized version of this algorithm for solving a perturbed inclusion $$y \in T(x),$$ where $y$ is a perturbation element near 0 and $T$ is a set-valued mapping. We will analyze the behavior of the convergent iterates generated by the algorithm and we will show that they inherit the regularity properties of $T$, and vice versa. We analyze the cases when the mapping $T$ is metrically regular (the inverse map has the Aubin property) and strongly regular (the inverse is locally a Lipschitz function). We will not assume any type of monotonicity.