In this paper we study a dynamical system of intervals moving on a metric graph with incommensurable edge lengths. This system can be generally viewed as a collection of congruent intervals moving around the network at unit speed, propagating along all respective incident edges whenever they pass through a vertex. In particular, we analyse the phenomenon of saturation: a state when the entire graph is covered by the moving intervals. Our main contributions are as follows: (1) we prove the existence of the finite moment of permanent saturation for any metric graph with incommensurable edge lengths and any positive length of the intervals; (2) we present an upper bound on the moment of permanent saturation. To show the validity of our results, we reduce the system of moving intervals to the system of dispersing moving points, for the analysis of which we mainly use methods from discrepancy theory and number theory, in particular Kronecker sequences and the famous Three Gap Theorem.