An action of a group $T$ on a set $X$ defines the action groupoidaction groupoid $T\times X \rightrightarrows X$ (If $T$ is a semigroup then $T\times X \rightrightarrows X$ is a category). Thinking of dynamical systems this way suggests that morphisms are functors between action groupoids. If there is a topology on $T$ and $X$ and the action is continuous, then the action groupoid is a topological groupoid. You may then take your morphisms to be continuous functors. However, this is not the best one can do. A better notion of morphism between topological groupoid is that of a bibundlebibundle also known as a ‘Hilsum–Skandalis map.’ If you go down this road you end up with bicategories since the composition of bibundles is associative only up to isomorphism (see this paperthis paper by Christopher Schommer-Pries for a nice discussion of the issues).
If your dynamical systems are vector fields on manifolds then the morphisms are smooth maps intertwining the vector fields. That is, if $X$ is a vector field on $M$, $Y$ a vector field on $N$ then a morphism from $(M,X)$ to $(N, Y)$ is a smooth map $f:M\to N$ with $Y \circ f = Df \circ X$.
There is also a large body of literature on the category of labelled transition systems and on the category of Petrie nets...
