  Motivated by the study of a certain family of classical geometric problems we
investigate the existence of multiplicative connections on proper Lie
groupoids. We show that one can always deform a given connection which is only
approximately multiplicative into a genuinely multiplicative connection. The
proof of this fact that we present here relies on a recursive averaging
technique. As an application we point out that the study of multiplicative
connections on general proper Lie groupoids reduces to the study of
longitudinal representations of regular groupoids. We regard our results as a
preliminary step towards the elaboration of an obstruction theory for
multiplicative connections.
