  We formulate Euler-Poincar\'e equations on the Lie group Aut(P) of
automorphisms of a principal bundle P. The corresponding flows are referred to
as EPAut flows. We mainly focus on geodesic flows associated to Lagrangians of
Kaluza-Klein type. In the special case of a trivial bundle P, we identify
geodesics on certain infinite-dimensional semidirect-product Lie groups that
emerge naturally from the construction. This approach leads naturally to a dual
pair structure containing \delta-like momentum map solutions that extend
previous results on geodesic flows on the diffeomorphism group (EPDiff). In the
second part, we consider incompressible flows on the Lie group of
volume-preserving automorphisms of a principal bundle. In this context, the
dual pair construction requires the definition of chromomorphism groups, i.e.
suitable Lie group extensions generalizing the quantomorphism group.
