The ‘river of time’ is actually the river of change (the only real evidence of Time flowing is change). Time (in this context) is merely a collective noun.
And we all change…continually. So we change, with everything around us changing.
But we all change independently. Change is reference frame specific.
So the big collective flow (Time) and the indivudual events (from quarks decaying, atoms vibrating, life breathing, earth spinning, universe expanding…depending on reference frame) have their own specific ‘flow’.