The DoF only changes if you compose the image the same on the two different sensors - that is, move closer to the subject with the full frame or use a different focal length so it fills the image just as it did on the crop. The change in depth of field comes from the closer subject distance that the full frame allows you to get as it has a bigger sensor. If you can't/don't want to get closer, you will get exactly the same DoF if you take a picture at 10m with a 50mm lens at f2 on a crop and full frame sensor, just you will get a wider view on the full frame.
This is sooo much easier to understand it you think of field of view. A full frame sensor has a wider field of view at a given focal length than the crop sensor. This allows you to compose your pictures in a different way (get closer/use different focal lengths for the same picture). Nothing more, nothing less...