Suppose that $\mathcal E=(E,\pi,M,F)$ and $\mathcal E^\prime=(E^\prime,\pi^\prime, M^\prime, F^\prime)$ are (locally trivial, smooth) fibre bundles (in this notation $E$ is the total space, $\pi$ is the projection, $M$ is the base space and $F$ is the standard fibre).
A morphism from $\mathcal E$ to $\mathcal E^\prime$ is a pair of smooth maps $(\phi,\phi_0)$ with $\phi:E\rightarrow E^\prime$ and $\phi_0:M\rightarrow M^\prime$ such that $\pi^\prime\circ\phi=\phi_0\circ\pi$. If $\phi$ is known, it uniquely determines $\phi_0$, thus it is sufficient to call $\phi$ the morphism.
Let now $\mathcal P=(P, \pi,M,G)$ and $\mathcal P^\prime=(P^\prime,\pi^\prime,M^\prime,G^\prime)$ be two principal fibre bundles.
In this case we can define two classes of principal morphisms.
In the first case $G=G^\prime$ and a morphism (in the sense of fibre bundles) $(\phi,\phi_0)$ is a principal morphism if the map is right-equivariant, i.e. for any $g\in G$ and $u\in P$ it is true that $\phi(ug)=\phi(u)g$. Principal bundles together with these morphisms form a category $\mathsf{PB}(G)$ of principal $G$-bundles.
In the second case, a morphism from $\mathcal P$ to $\mathcal P^\prime$ is a triple $(\phi,\phi_0,\varphi)$, where the first two elements form a morphism of fibre bundles, $\varphi:G\rightarrow G^\prime$ is a Lie group homomorphism, and the fibred morphism satisfies the generalized equivariance condition $\phi(ug)=\phi(u)\varphi(g)$ for $g\in G$ and $u\in P$. Principal fibre bundles with these morphisms form the category $\mathsf{PB}$ of principal bundles.
Let now $\mathcal E=(E,\pi,M,F,G)$ and $\mathcal E^\prime=(E^\prime,\pi^\prime, M^\prime, F^\prime, G^\prime)$ be two fibre bundles with structure group. One may define fibre bundle morphisms between them as before, but these morphisms will not respect the $G$-structures.
As it is well-known, every fibre bundle with structure group is associated to a unique principal fibre bundle. I know that principal morphisms of principal fibre bundles induce morphisms of the associated bundles. I don't remember the exact details but I know where I can check it.
On the other hand, suppose that we do not want to define principal bundles or understand fibre bundles with structure group through the associated bundle construction. We just want to deal with fibre bundles with structure group without principal bundles.
How does one define the $G$-structure-preserving morphisms between fibre bundles with structure groups then?