  The article is devoted to the study of mappings with finite distortion in
metric spaces. Analogues of results relating to equicontinuity and normality of
families of quasiregular mappings are obtained. It is proved that the indicated
families are equicontinuous if the characteristic of the mappings has a finite
mean oscillation at each inner point, and the maps omit a certain fixed
continuum. An equicontinuity of generalized quasiisometries on Riemannian
manifolds is also obtained.
