virtual chain conditions in groups

In group theory, it's often very useful to know whether a family of subgroups (eg normal subgroups, Zariski-closed subgroups, ...) satisfies an ascending chain condition or a descending chain condition (that is, all ascending/descending chains in this family are finite). What I'm interested in is weaker 'virtual' chain conditions: a virtual DCC would be that given a sequence $G_1 > G_2 > \dots$ of subgroups (of some special kind) such that $G_{i+1}$ has infinite index in $G_i$ for all $i$, then the sequence must terminate. One can define virtual ACCs similarly.

Does anyone know of work that has been done on conditions of this kind, either showing that a family of subgroups satisfies the conditions or deriving consequences from them? References for analogous conditions in other algebraic contexts would also be interesting.

The virtual DCC doesn't seem so different from the notion of Krull dimension $1$ that I explained in answer to this Different definitions of the dimension of an algebra question.