We explore new representations for lattice gauge theories with fermions, where the space-time lattice is divided into dynamically fluctuating regions, inside which different types of degrees of freedom are used in the path integral. The first kind of regions is a union of so-called bags, in which the dynamics is described by the free propagation of composite degrees of freedom of the original fermions. In the second region, called complementary domain, configurations of the remaining interacting degrees of freedom are used to describe the dynamics. We work
out the bag representation for the gauge groups SU(2) and SU(3) and address the nature of the strong coupling
effective degrees of freedom, which are fermions for SU(3) and bosons for SU(2). We discuss first steps towards a
numerical simulation of the bag representations.