We investigate modifications of the selection principles for various kinds of convergences introduced by L. Bukovsk\'y and J. \v Supina [9]. We show that if we restrict our selection principles to continuous functions then they split just to two equivalence classes. Considering other families of limit functions we obtain our main result saying that selecting ``from every sequence of a sequence of sequences of functions'' or just ``from infinitely many sequences'' can produce properties which can be consistently distinguished. Moreover, we show that our selection principles are characterizations of some properties of a topological space, e.g. hereditarily Hurewicz space or hereditarily $S_1(\Gamma,\Gamma)$-space.