Recently I've read that if fundamental theory with fermions contains global current anomalies, then in effective field theories (when we integrate out some fermions) these anomalies can be described in following way: suppose we make global symmetry $G$ associated with current local, i.e., we introduce some background gauge field $B$ which is transformed under adjoint representation of $G$. After integration out of some fermions Wess-Zumino term arises. We calculate its variation under local transformation of $G$, and by wondering to have zero variation we introduce anomalous part of local current which has anomalous conservation law.

But why this method is correct? Does introduction of new background field leave theory consistent (for example, new mixed anomalies arise)? And how to make calculations with fictive background field?

An edit. It seems that background gauge fields corresponds to the vector fields which arise in effective theory. For example, in QCD these fields are associated with vector mesons. But I don't understand why treating of mesons as gauge fields promote correct effective interactions description.