Depth of field - does sensor size affect it beyond lens multiplier effect?

I am fully aware of depth of field as a function of aperature and magnification power (ie low depth of field with higher power with wide open aperature). What I don't understand is if there any significance of the sensor size beyond the multiplication factor.

In other words, if I had two camera bodies, one a crop body with 1.6x sensor, and another with full frame. If I shoot full frame using 2.8 using a 320mm tele, would it have the same depth of field as a 1.6 crop shooting f2.8 using a 200mm (which is 320mm with the 1.6x magnifier)?

Or is there something else about sensor size that would give the full frame shooting f2.8 at 320mm less or more depth of field? If so, would the full frame give you less depth of field (more blurry background) or more depth of field (less blurry background)?