How are sources described in gauge theory?

Because gauge invariance requires covariant "conservation" $$ 0=nabla_mu J^muequiv partial_mu J^mu +[A_mu, J^mu], $$ the sources $J^mu$ in non-abelian theory cannot be $c$-numbers. Instead they are Wilson Loops. You can also make them classical but with internal dynamics, as in the Wong equations.

Notice that the "charge" of a source in Yang-Mills theory is the representation of the gauge group in which the Wilson line belongs. For example in the SU(3) colour theory quarks have charge "${bf 3}$" because they have 3-d representation and gluons have charge "${bf 8}$" because they are in the 8-dimensional adjoint representation. "Addition" of the charges is the Clebsch-Gordan decomposition of the tensor product of the representations such as $$ {bf 8}times {bf 8}= {bf 1}+ {bf 8}+{bf 8}+ {bf 10}+overline {bf 10}+{bf 27}. $$ In particular gauge "charge" must come in discrete lumps. You cannot have a smooth charge distribution.

Working out the effect of the sources on the gauge bundle is non trivial. For an abelian theory they just induce a holonomy. For non-abelian theories the result is highly non-trivial. The simplest case that I know of results in the Knizhnik-Zamolodchikov equations of conformal field theory, and its spacetime interpretation as Witten's knot invariants.