The evolution and dispersion of intermediate water masses in the ocean interior is studied. To this purpose, an empirical statistical model of Lagrangian tracers at a constant depth level is developed. The model follows the transfer operator based on 10-day deep displacements of Argo floats at; similar to 1000m depth. An asymptotic analysis of the model shows the existence of 10 principal stationary points (the 10 locations attract asymptotically 97% of the tracers). It takes; 1000 years to reach this asymptotic regime relevant for estimating the stationary points. For Lagrangian floats, the concept of attractor needs to be generalized in a statistical sense (versus deterministic), except for a few places in the ocean. In this new framework, a tracer has a likelihood to reach the stationary points, rather than a certainty to reach a single stationary point. The empirical statistical model is used to describe the fate of three intermediate water masses: North Pacific Intermediate Water (NPIW), Mediterranean Water (MW), and Antarctic Intermediate Water (AAIW). These experiments show a dramatic difference in the long-time behavior of NPIW, MW, and AAIW. In the permanent regime, the NPIW concentrates locally (in the Kuroshio recirculation) and the MW remains mainly regional (concentrated in the subtropical gyre of the North Atlantic), whereas the AAIW spreads globally (well mixed throughout the entire ocean).