Consider the following model for percolation in an infinite graph: each vertex has a certain region (set of vertices) associated with it, which at the beginning contains only the vertex itself, and grows with time. Each vertex holds a random Poisson clock, all clocks independent, and at each tick of a clock of a vertex $v$, $v$'s region is updated to the current region united with the regions of all neighbors of the $v$ (which might have grown by that time).

The question about this model is whether there exists a vertex whose region percolates = becomes infinite at a finite time with positive probability.

Does this model have a standard name and/or has been studied before?