Let G be a locally compact group, let K be a closed subgroup of G, and let H be a group of automorphisms of G such that h(K)=K for all hinH. When is the action of H on G/K a small invariant neighbourhoods (SIN) action, i.e. when is there a basis of neighbourhoods of the trivial coset consisting of H-invariant sets? In general, the SIN property is a strong restriction, but when G is totally disconnected and H is compactly generated, it turns out to be equivalent to the seemingly weaker condition that the action of H on G/K is distal on some neighbourhood of the trivial coset. (The analogous statement is false in the connected case: compact nilmanifolds give rise to counterexamples.) This has some general consequences for the structure of t.d.l.c. groups: for example, given any compact subset X of a t.d.l.c. group G, there is an open subgroup containing X that is the unique smallest such up to finite index.