Let $H$ be a group acting by homeomorphisms on a Hausdorff space $X$.  Say the action is *distal* if for all $(x,y) \in X \times X$, if the set $\{(hx,hy) \mid h \in H\}$ accumulates at a diagonal point $(z,z)$, then $x=y$.

Now let $G$ be a locally compact group and let $H$ be a group of automorphisms of $G$.  Let $K$ and $L$ be $H$-invariant closed subgroups of $G$ such that $L \le K$.  The coset spaces $G/K$, $K/L$ and $G/L$ then carry the quotient topology.

1. To show $H$ acts distally on $G/K$, is it enough to show that no $H$-orbit accumulates at the trivial coset?

2. If $H$ acts distally on $G/K$ and on $K/L$, does it follow that $H$ acts distally on $G/L$?