If the dot is animating with an expression already you'll just need to add a time offset to expression. So if your original expression was something like this:
freq = 1;
t = Math.cos(freq * time * 2 * Math.PI);
linear(t, -1, 1, value, 0);
Then you could add an offset of .1 seconds like this:
offset = .1;
freq = 1;
t = Math.cos(freq * (time - offset) * 2 * Math.PI);
linear(t, -1, 1, value, 0);
Alternatively you could make this offset based on the inPoint of the layer: