I'd like to understand the following random graph process. I'm not sure if it's difficult or straightforward, so apologies if this is below the level of mathoverflow, but I've gotten no response on math.stackexchange. Any pointers to useful references will also be appreciated.

The process works as follows: we have a set of vertices, $V$, and a set of colors, $C$, and we are going to build a digraph such that each edge has a color and each vertex has at most one outgoing edge of each color. Pick a starting vertex $v_0$, at random. At each step, whatever the current vertex is, say $v_i$, pick a color $c$ at random and a vertex $v_{i+1}$ uniformly among those vertices which do not have an outgoing edge of color $c$ to a vertex other than $v_i$. Then if there isn't already, add an edge of color $c$ from $v_{i+1}$ to $v_i$ and make $v_{i+1}$ the new current vertex.

For a given set of vertices and colors and number of steps, each allowable graph has some probability of occuring, and I'd like to understand in particular how far this probability is from being uniform on allowable graphs.