  We provide quantitative and abstract strong convergence results for sequences
from a compact metric space satisfying a certain form of generalized Fej\'er
monotonicity where (1) the metric can be replaced by a much more general type
of function measuring distances (including, in particular, certain Bregman
distances), (2) these distance functions are allowed to vary along the
iteration and (3) full Fej\'er monotonicity is relaxed to a certain partial
variant. To that end, the paper provides explicit and effective rates of
metastability and even rates of convergence for such sequences, the latter
under a regularity assumption that generalizes the notion of metric regularity
introduced by Kohlenbach, L\'opez-Acedo and Nicolae, itself an abstract
generalization of many regularity notions from the literature of nonlinear
analysis.
  In the second part of the paper, we apply the abstract quantitative results
established in the first part to two algorithms: one algorithm for
approximating zeros of maximally monotone and maximally $\rho$-comonotone
operators in Hilbert spaces (in the sense of Combettes and Pennanen as well as
Bauschke, Moursi and Wang) that incorporates inertia terms every other term and
another algorithm for approximating zeros of monotone operators in Banach
spaces (in the sense of Browder) that is only Fej\'er monotone w.r.t. a certain
Bregman distance.
