  In this work, we develop a Lagrangian reduction theory for covariant field
theories with gauge symmetries. These symmetries are modeled by a Lie group
fiber bundle acting fiberwisely on a configuration bundle. In order to reduce
the variational principle, we utilize generalized principal connections, a type
of Ehresmann connections that are equivariant by the fiberwise action. After
obtaining the reduced equations, we give the reconstruction condition and we
relate the vertical reduced equation with the Noether theorem. Lastly, we
illustrate the theory with several examples, including the classical case
(Lagrange-Poincar\'e reduction), Electromagnetism, symmetry-breaking and
non-Abelian gauge theories.
