maybe the clearest way to explain this is: if you compare it to a cropped (aps-c) sensor, shooting the same thing from the same point, to get the same result (framing) on full frame you need a longer focal length; as a matter of fact, longer lenses give a shallower depth of field.
edit(more accurate, more annoying): so it's uncorrect and misleading to say a bigger sensor leads to a shallower depth of field, at least directly, no way this could actually happen.
Depth of field really depends on focal lenght and f-number ONLY; on every possible sensor (or film) the same lens (same settings) will give the same depth of field.
If we assume every camera we put this given lens on has the right mount to fit it and the same distance between the lens and the focal plane to get focus right, all we get comparing all possible sensor/film formats is different cropped parts of the same projected image circle.
UPDATE: Here's an update to tell my commenters (and everybody interested) why I didn't account for CoC in relation to sensor size in my answer above. Most people who may read this already know probably more than me about this subject and may find stupid to read something they already know; keep in mind I'm just trying to put this in perspective.
For those who don't know, a Circle of Confusion is one among infinite dots of light a lens projects on its focal plane, in our case the part of a camera where a sensor or some film is. The closest this dot gets to a point (which has no size, but our dot is never a point because lenses are not perfect), the sharper the image (in that area), and vice versa. (A rough definition just to introduce newcomers to the subject, experts please don't waste time criticizing this).
I think we can agree the maximum permissible CoC is a value used to determine with mathematical certainty what is in focus and what is not; it serves the purpose of "drawing a line" (as actually there is a smooth transition between in focus and not in focus), and it may well be adjusted according to the print size you want to achieve, as a larger print would make this transition more evident and, at some level, one would notice that some areas, appearing to be in focus looking at a smaller print of the same image, are actually out of focus.
Applying formulas in which one element is the CoC to determine DOF is an analytic process, so it's a way to understand what's going on, not changing things (i.e. the image a lens is projecting on a focal plane). The fact that CoCs exist and that one needs to decide what's the right CoC size to produce a sharp image of a given size is not changing the way a lens works across different sensors/film formats.
If you want to get a large print, I understand it may be needed to consider a different acceptable sharpness.
In my answer above I assume (and I declared I do) the focal plane is always the same, that's what's happening in modern DSLRs we're talking about: the different rendition of a lens on different bodies is just a matter of cropping.
an ugly oversimplified drawing sometimes helps
making two print of the same size from the two different sensors will give varying results depending on the resolution of the two sensors.
if we assume pixel density is the same for both sensors and we print at a fixed resolution, the prints from the smaller sensor will look exactly like the larger ones, just cropped.
if again we assume pixel density is the same and we print at a fixed size, the prints from the smaller sensor will look like enlarged crops of the larger ones, at a lower quality.
if we simply and more correctly to the purpose of our analysis assume both have enough pixels to print a fixed size we want without noticeable loss in quality, what we get in prints from the smaller sensor is like a cropped enlargement of the print from the bigger sensor, so we can indeed see a difference in depth of field, that is, we see more detail so it's easier to spot slightly out-of-focus areas.
This we could have accounted for if we decided to put a larger CoC in the formula when, while shooting, we calculated the DOF. Who's so rough a photographer not to do so? ;-)
However, I'm not really mocking anyone. I'm just saying, long story short: if you do big prints you might want to squeeze a bit more DOF out of your aperture selecting a higher f-number, so some part of the image around the line separating out of focus and in focus will be sharp enough to be considered fully in focus also in enlargements. That's it.
The same lens with the same aperture settings (at the same focal length if it's a zoom) will always give the same result. CoC is not a physical variable like those I just mentioned which really changes light coming out of the lens and into you camera, it's a parameter used to determine mathematically if something is in focus.
You can't say DOF is a function of sensor size (among others) because larger sensors are used for large prints and in large prints you see out of focus areas that in small prints you don't see. First because to call this an indirect relation is an euphemisms, then because this is accounting for detail at the expenses of sheer exactness. Maybe I'm missing on... more than something.