Ever since Furstenberg proved his multiple recurrence theorem, the limiting behaviour of multiple ergodic averages along various sequences has been an important area of investigation in ergodic theory. In this talk, I will discuss averages along arithmetic progressions in which the differences are elements of a fixed integer sequence. Specifically, I will give necessary and sufficient conditions under which averages of fixed length of the aforementioned form have the same limit as averages along arithmetic progressions of the same length. The result relies on a higher-order version of the degree lowering argument, which is of independent interest. The talk is based on a joint work with Nikos Frantzikinakis.