Suppose we have a principal $G$-bundle $P\xrightarrow{\pi} M$, and we want to consider a classical gauge field theory (with a field Lagrangian) on $M$ for this bundle, in the physics sense.
In the physics literature they seem to use a "gauge field" on $M$, which seems to me to be $s^*A$ where $s$ is a (global? local?) section of $\pi$ and $A$ is a principal bundle connection (as defined here).
My question is, do we assume $\pi$ admits a global section $s$ then work with $s^*A$ as the gauge field? In which case the principal bundle would be globally trivial?
Furthermore, in the above case wouldn't the theory depend in some nontrivial way on the choice of section (e.g. its homotopy class)?
Or, do we consider $M$ to be covered by a collection of local sections, and we assume that the Lagrangian/action is invariant under the "transition maps" between $s_i^*A$ and $s_j^*A$?