Some covering properties of topological spaces which are defined in terms of possibility to choose from a given sequence of covers (of some kind) a cover of the same or a different sort are considered. In particular we are interested in preservation of such properties in the preimage direction under several sorts of continuous mappings. The properties include the classical concepts: the Menger property, Rothberger's property and so on. For example, it is shown that the Menger property (in all finite powers) is an inverse invariant of closed irreducible finite-to-one mappings.