Let $G$ be a path-connected topological group that acts on the path connected space $X$. Let $\alpha$ be a loop in $G$ based at the identity, and let $x, y$ be points in $X$. I want to show that $\alpha(x)$, which is a $x$-based loop in $X$ is homotopic to the constant loop at $x$ iff $\alpha(y)$ is homotopic to the constant loop at $y$.
If $G$ acted transitively on $X$, then I think I can conclude that the loop $\alpha$ in $G$ contracts to the constant loop at the identity (since for any image of the square in the homotopy between $\alpha(x)$ and $c_x$, I can choose the group element that sends $x$ to that point), and then I can use that homotopy in $G$ to construct a homotopy between $\alpha(y)$ and $c_y$. But is it the case that $G$ must act transitively on $X$. And if not, how can I prove the statement in general?
Let $\tau : [0,1] \to X$ be a path from $\tau(0)=x$ to $\tau(1)=y$. Then, for any loop $\alpha : [0,1] \to G$, $\alpha(0) = \alpha(1) = 1_G$, we have a homotopy $H : [0,1] \times [0,1] \to X$ given by $H(s,t) = \alpha(s) \cdot \tau(t)$. It goes from $H(s,0) = \alpha(s) \cdot x$ to $H(s,1) = \alpha(s) \cdot y$ and goes through loops since $H(0,t) = H(1,t) = \tau(t)$ for any $t$.
This shows that the loops $\alpha(x)$ and $\alpha(y)$ are freely homotopic (i.e., homotopic through maps that don't preserve basepoints). Free homotopy classes of loops in a path connected space correspond to conjugacy classes in the fundamental group, and the constant loop is its own conjugacy class, so $\alpha(x)$ is null iff and only if $\alpha(y)$ is.
In more detail: the homotopy $H$ can be reinterpreted as a homotopy relative to the endpoints between the top side, $H(s,0) = \alpha(s) \cdot x$ and the concatenation of the three other sides, $H(1, 1-\ast) \circ H(\ast, 1) \circ H(0,\ast) = \tau^{-1} \circ \alpha(y) \circ \tau$. If $\alpha(x) \sim c_x$, we get $\tau^{-1} \circ \alpha(y) \circ \tau \sim \alpha(x) \sim c_x$ and thus $\alpha(y) \sim \tau \circ c_x \circ \tau^{-1} \sim c_y$ where $\sim$ is homotopy relative to endpoints and $c_x, c_y$ are constant loops.