In this paper, we consider composition principles for generalized almost
periodic functions. We prove several new composition principles for the classes
of (asymptotically) Stepanov $p$-almost periodic functions and (asymptotically,
equi-)Weyl $p$-almost periodic functions, where $1\leq p<\infty ,$ and explain
how we can use some of them in the qualitative analysis of solutions for
certain classes of abstract semilinear Cauchy inclusions in Banach spaces