There are a couple of ways to define an action of $\pi_1(X)$ on $\pi_n(X)$. When $n = 1$, there is the natural action via conjugation of loops. However, the picture seems to blur a bit when looking at the action on higher $\pi_n$. All of them have the flavor of the conjugation map, but are more geometric than algebraic, and in some cases work is needed to show the map is well defined. Here are a couple I have seen:

There is a homotopy equivalence $f : S^n \to S^n \vee I$. taking the basepoint of $S^n$ to the endpoint of the unit interval "far away" from $S^n$. Given a path $\alpha$ from $x_0$ to $x_1$, one can get a basepoint changing homomorphism $\pi_n(X,x_0) \to \pi_n(X,x_1)$ by taking $g : S^n \to X$ and mapping it to $(g \vee \alpha) \circ f$. If $\alpha$ is a loop this gives an action of $\pi_1$

Another way to proceed may be to look at elements of $\pi_n(X,x_0)$ as homotopy classes of maps $I^n \to X$ that send $\partial I^n$ to $x_0$. Then a base change homomorphism could be obtained by using a path $\alpha$ to define a map $I^n \cup (\partial I^n \times I) \to X$, which can be filled in to a map $I^{n+1} \to X$. Then the action would be to take the face opposite the original $I^n \subset I^{n+1}$.

These both define the same standard action of $\pi_1$ on $\pi_n$, but lose the algebraic flavor of the group action and instead have this stronger geometric feel, which can make working with the action a bit cumbersome. Are there other ways of looking at this action that are more algebraic?

Perhaps, can something be done wherein $\pi_0(Y)$ acts on $\pi_n(Y)$, where $Y$ is some sufficiently nice space like $\Omega X$, and does this coincide with the above defined actions? Is this a useful way of viewing the action?