Let $f,g \colon X \to Y$ two cellular maps between (say) finite CW complexes such that $f\sim g$ via the homotopy $H \colon X \times I \to Y$.
Are there any results that permits to modify the given homotopy $H$ such that for every $t \in I$, the map $H_t \colon X \to Y$ is cellular?
I read that there is a notion of cellular homotopy, but the definition doesn't imply the highlighted property. In fact an homotopy is said ti be cellular, if it is a cellular map from $X\times I \to Y$ seen as a map of CW complexes.
This is really a comment rather than an answer, but a bit too long for a comment.
There is a general notion of filtered space $X_*$, and of maps and homotopies between them, see Nonabelian Algebraic Topology. Of course a CW-complex with its skeletal filtration is a key example, In this book a homotopy such as you describe is called a thin homotopy, and these play a special role in the theory and applications. For example there are rather obvious compositions in various directions, generalising the composition of paths, of filtered maps $I^n_* \to X_*$, where $I^n_*$ is the $n$-cube with its usual skeletal filtration; a key result is that these compositions are inherited by thin homotopy classes, to give strict $n$-fold groupoids, generalising the fundamental groupoid on a set of base points.