On the global behavior of mappings and the correspondence of boundaries

We consider families of mappings with moduli inequalities, having different definition domains. Under some additional assumptions we have proved that such families are uniformly equicontinuous. We have considered four main cases: when mappings are homeomorphisms and corresponding domains have simple geometry; when similar mappings have branch points; when domains with complex geometry are considered, but mappings still are homeomorphisms; and when similar mappings have branch points. Sequences of domains are generally assumed to converge to a kernel, and the characteristics of the mappings must satisfy certain conditions on their growth. In some of the four cases mentioned above, we also described properties of the limit mapping. We also obtained the correspondence of the boundary points of the kernel to the boundary points, and the inner points to the inner points.