Gradient-like diffeomorphisms and periodic vector fields

A class of gradient-like nonautonomous vector fields (NVFs) on a smooth closed manifold $M$ is studied and the following problems are solved: 1) can a gradient-like NVF be constructed by means of the nonautonomous suspension over a diffeomorphism of this manifold, and if so, under what conditions on the diffeomorphism? 2) let a diffeomorphism $f$ be gradient-like (see the definition in the text) and diffeotopic to the identity map $\mathrm{id}_M$, when the NVF obtained by means of the nonautonomous suspension over $f$ be gradient-like? Necessary and sufficient conditions to this have been found in the paper. All these questions arise, when studying NVFs on $M$ admitting the uniform classification and a description via combinatorial type invariants.