IMPORTANT NOTE: I just realized that I accidentally posted this to stackexchange rather than MathOverflow; the latter seems more suitable. So, anyone interested in following up should go to https://mathoverflow.net/questions/262508/diamond-lemma-for-edge-colored-directed-graphs . Sorry for creating confusion!
Suppose we have a directed graph with colored edges such that (1) every vertex has at most one outgoing edge of each color, and (2) for any path of length two, say from $u$ to $v$ to $w$, there exists a vertex $v'$ and a directed path from $u$ to $v'$ to $w$ such that edge $(u,v')$ has the same color as edge $(v,w)$ and edge $(v',w)$ has the same color as edge $(u,v)$. Then a standard argument (the "Jordan-Holder argument") shows that, for any starting vertex $s$, EITHER there is no terminating path from $s$ (where a terminating path is one whose final vertex $t$ has no outgoing edges) OR every maximal path from $s$ terminates at a path-independent vertex $t$ and moreover the number of times each color gets used is path-independent.
Can anyone provide a citation for this lemma or something like it? (It occurs in a more specialized form in the theory of abelian sandpiles, and again in the theory of rewrite systems, and again in group theory, but it's really just a general-purpose graph-theory lemma.)
NOTE: The statement is incorrect. One fix is to add condition (3): for all $v$, if we have distinct edges $(v,w)$ and $(v,w')$, there exists $x$ and edges $(w,x)$ and $(w',x)$ with the same color as $(v,w')$ and $(v,w)$ respectively.