It is known that Dobrushin's ergodicity coefficient is one of the effective tools in the investigations of limiting behavior of Markov processes. Several interesting properties of the ergodicity coefficient of a positive mapping defined on base norm spaces have been studied. In this paper, we consider uniformly mean ergodic and asymptotically stable Markov operators on such spaces. In terms of the ergodicity coefficient, we establish uniform mean ergodicity criterion. Moreover, we develop the perturbation theory for uniformly asymptotically stable Markov chains on base norm spaces. In particularly, main results open new perspectives in the perturbation theory for quantum Markov processes defined on von Neumann algebras.