Let $G$ be a topological group, let $K$ be a closed cocompact subgroup (i.e. the coset space $G/K$ is compact in the quotient topology) and let $g \in G$. Is there a sequence of positive powers $g^{i_n}$ of $g$ such that $g^{i_n}K$ converges to $K$ in the coset space $G/K$?

If the answer is `no' in general, what if $G$ is totally disconnected and locally compact? (For the application, I'd be happy if I could at least get powers of $g$ to land in $UKV$ for any pair of identity neighbourhoods $U$ and $V$.)