I understand for a pair of smooth vector bundles $E$ and $F$ over a smooth manifold $M$ it makes sense to talk about the differential operators between the section spaces of $E$ and of $F$. The definition of a differential operator involves picking local sections of $E$ and $F$. Since locally both of them are vector spaces, local sections can be thought of as several smooth functions over $M$.
Question
Does it make sense to talk about differential operators between two principle $G$-bundles? Or even for general fibre bundles? I can't see a direct generalization (if any) since we can't interpret the local sections as several smooth functions anymore.