(Joint work with Dan Berwick-Evans and with Stafan Klajbor-Goderich.)
I will argue that it is useful to think of of invariant vector fields on a manifold with a group action as objects of a groupoid. Isomorphic vector fields are similar dynamically. For example they have the same relative equilibria with the same stability.
More generally multiplicative vector fields on a Lie groupoid are also objects of a groupoid, which behaves well under Morita equivalence. In both cases the groupoids of vector fields are Lie 2-algebras.