Consider a pair of principal bundles $P \to M$ and $P' \to M'$ with groups $G$ and $G'$, respectively. A morphism from $P$ to $P'$ is a pair $(\Phi, \phi)$ where $\phi: G \to G'$ is a Lie group homomorphism and $\Phi: P \to P'$ is a fibre bundle map which is equivariant with respect to $\phi$ in the sense that $\Phi(u.g)= \Phi(u).\phi(g)$. This gives us a category $\mathsf{Prin}$ of principal bundles and their homomorphisms. If we require that $G = G'$ and $\phi= \mathrm{id}_G$ then we get the subcategory $\mathrm{Prin}G$ of $G$-principal bundles.

Now, we know that the construction of associated bundles is a bifunctor $\mathsf{Prin} G \times G\mathsf{Man} \to \mathsf{Bund}$, where $G\mathsf{Man}$ is the category of manifolds with left $G$-action and $\mathsf{Bund}$ is the category of fibre bundles.

Now, suppose that we pick a morphism of principal bundles $(\Phi, \phi)$ as above, where $G$ and $G'$ might differ. Then what is the relationship between the corresponding associated bundle functors? Do we get a natural transformation between the functors, for example $P' \times_{G'} (-) \Rightarrow P \times_{G}\phi^*(-)$, where $\phi^*: G'\mathsf{Man} \to G\mathsf{Man}$ is the restriction functor? If so, under what conditions would this be a natural isomorphism?

For a concrete example, suppose $(\Phi, \phi)$ is a morphism from a $\mathrm{Spin} (n)$ principal bundle $P$ to an $\mathrm{SO}(n)$-principal bundle $P'$, where $\phi: \mathrm{Spin}(n) \twoheadrightarrow \mathrm{SO}(n)$ is the universal covering space. Typically, $P'$ will often be the orthonormal frame bundle of an oriented semi-Riemannian manifold.

My go-to reference for these sort of things is Natural Operations in Differential Geometry by Kolar, Michor, and Slovak, but I couldn't find an answer there. I have this question tagged as spin geometry because I figure people working in this field have thought about this sort of question before.