Let $G$ be a compact connected Lie group acting transitively and smoothly on another compact Lie group $K$. Let $d$ be the distance in $K$ that is not $G$-invariant. Is there a constant $C$ such that $d(gx,gy)\leq Cd(x,y)$ for every $g\in G$ and $x,y\in K$? Moreover, the action is not isometric.

The path I tried by MVT is leading me to bound $ \lVert dg_{x}\rVert_{\operatorname{op}}$ for $g\in G$ and for $x\in K$:

Let $\gamma$ be a geodesic connecting $x$ to $y$. For each $g\in G$, $g\gamma$ is a path connecting $gx$ to $gy$, so $$d(gx,gy)\leq \int_0^1|(g\gamma)'(t)|dt=\int_0^1|dg_{\gamma(t)}\gamma'(t)|dt\leq\int_0^1\lVert dg_{\gamma(t)}\rVert_{\operatorname{op}}|\gamma'(t)|dt$$

It would suffice to show that $(g,x)\mapsto dg_{x}$ is continuous in the topology of the operator norm. But it is proving to be harder than I thought.