Let's have a look on the empty FRW-Universe. Though being empty it's metric is expanding, regardless whether or not one imagines massless testparticles, whose distances are increasing. The particles are helpful to have a vision of the "flow", but they don't create it.It is only meaningful to state that it is matter/energy that is "flowing" in the models of an expanding universe, and that the statement that "empty space" is "flowing" and expanding in all directions has no physical meaning. Empty space doesn't flow and we do not attempt to measure to measure any characteristic of empty space's nothingness to prove that this nothingness is the source of some physically relevant effect.ing.
One can moreover imagine a large non-expanding (due to internal forces) ruler. Then you can visualize the increasing distances between the particles travelling along the ruler. The volume of the cube defined by the ruler isn't interpretable, but the increasing distances are (expansion vs. motion), as pointed out several times by Chalnoth. Admittedly in the case of the empty universe it seems much easier to interpret the recession of "galaxies" as being due to a motion, because in the Milne model, which is equivalent to the empty FRW model, the redshift is purely a Doppler shift.