Summary: We consider a gossip-based distributed stochastic approximation scheme wherein processors situated at the nodes of a connected graph perform stochastic approximation algorithms, modified further by an additive interaction term equal to a weighted average of iterates at neighboring nodes along the lines of ``gossip'' algorithms. We allow these averaging weights to be modulated by the iterates themselves. The main result is a Benaim-type meta-theorem characterizing the possible asymptotic behavior in terms of a limiting o.d.e. In particular, this ensures ``consensus,'' which we further strengthen to a form of ``dynamic consensus'' which implies that they asymptotically track a single common trajectory belonging to an internally chain transitive invariant set of a common o.d.e. that we characterize. We also consider a situation where this averaging is replaced by a fully nonlinear operation and extend the results to this case, which in particular allows us to handle certain projection schemes.