Very much so. There are a number of small industries centred around studying equivalence classes of knot diagrams generated by a set of moves.
- The study of claspers. For example, $C_k$-moves are a special type of clasper surgeries. MathSciNet indicates 123 citations for Habiro's fundamental paper Claspers and finite type invariants of links, providing some coarse measure of the vitality of the topic.
- Replacing one rational tangle in a knot diagram by another generates an equivalence relation which has been deeply studied using quandles. See e.g. J. Przytycki's introductory lectures.
- Dehn surgery, where the surgery curve is required to belong to some specified part of a knot group or link group (in the kernel of its representation to some fixed group, for instance) generates equivalence relations on knot diagrams modulo combinatorial "twisting" moves, which have been studied by Cochran-Orr-Gerges, and (excuse the self promotion) by myself and Andrew Kricker, and by Litherland and Wallace. The techniques for studying these equivalence relations have been topological rather than combinatorial.
- There are a number of setting in which one allows Reidemeister moves plus some crossing changes, but not others. In the theory of finite type invariant, one fixes a some crossings (considers them in resolutions of "double points"), and allows crossing changes away from them. The equivalence classes are detected by the finite-type invariant of type the number of "fixed" crossings. In a similar-sounding vein, a free virtual knot is a virtual knot where we allow crossing changes away from virtual crossings. They have a rich theory- see e.g. this Manturov paper.