Not really an answer, but a comment about the related question:
Can one reach every cluster in that way ?
It is rather easy to check (using sagemath or Keller's applet) that for the cyclic quiver in type $D_n$, not every cluster can be reached in that way. For example, only 13 among the 14 clusters in type $D_3$ are reached.
A conjecture for the number of reached clusters in type $D_n$ can be found in this OEIS sequence.