Under what circumstances can it be concluded that if two items from the smooth category are related by a topological relationship, then they are also smoothly related in the corresponding way? For instance, it is true that two smooth maps between manifolds without boundary $M$ and $N$ if $N$ is compact always satisfy that if $f, g$ map $M$ into $N$ and $f, g$ are homotopic, then they are also smoothly homotopic. Here the items are f and g, while the relation is homotopy which upgrades to smooth homotopy. (Although a different homotopy may be required to mediate this.) I'm not sure if this claim persists when boundaries are allowed, or when N is allowed to be not compact. Please let me know if boundaries are allowed in your claims/proofs, in addition to whatever other conditions are required.
If $M$ and $N$ are homotopy equivalent, what can one say? What if one is a deformation retract of the other? What if they are homeomorphic?
More esoterically, what if one is a (continuous) covering space for the other? Feel free to add to this if anything hits you, and please let me know if there is a general technique that catches all questions of this type. Thanks.