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.