First recall that DoF is not a distance that comes solely from optical theory (like the distance to the image for a certain object location); it is a description of the range of object distances over which a certain arbitrarily-adopted criterion of "blurred-ness" is not exceeded, with the camera focused at a certain distance.
Often, for consistency, that criterion is a blur circle whose diameter is a certain fraction of the overall image size (maybe its diagonal size). This relates to a consistent diameter blur circle on an image produced at a consistent size (such as a consistent size print).
That having been said, when we compare the DoF attained for two values of some attribute of the camera (such as sensor size), "all other factors being equal", we must adopt and announce what we mean by "all other factors being equal".
Here is one set of such that we might adopt:
a. Focal length such that the field of view is the same in both setups. (Aha!)
b. Camera focused at the same distance. (Of course.)
c. Same f-number. (Of course.)
d. Criterion for "acceptable" blurring the same in terms of blur circle diameter as a fraction of the image size (diagonal will do).
Now, if we are comparing two cameras, with "B" having an image size 2x that of "A":
1. Under rule a, we must use a lens of twice the focal length in B as in A.
2. Under rule c, our acceptable diameter of blur circle is twice for B as it is for A.
Now, with regard to point 2, that means that we are more tolerant of blur in B than in A. Thus this consideration alone would lead to a greater depth of field for B than A.
This works essentially proportionally to the blur circle diameter criterion, and thus to image size (sensor size).
Remember, depth of field is not an creature of optical theory alone. It is a creature of what amount of blurring we consider "acceptable". If we increase the amount we consider "acceptable", then our focus distance can be more "off" and we still consider the result acceptable.
Now, to point 1. Because of the optical situation involved (and I will not attempt to describe this in detail here), for a greater focal length the "incorrectness" in focus distance to cause a certain diameter blur circle is less. Thus this consideration alone would lead to a lesser depth of field for B than A.
This works essentially proportionally to the square of the focal length, and thus (because of rule a) to the square of the image size (sensor size).
So as we move from A to B, we find that issue 2 gives an increase in the depth of field, and issue 1 gives a larger decrease in the depth of field (because it varies inversely as the square of the image size).
Thus, the overall effect is that (under the rules stated above), for an increase in sensor size we have a net decrease in depth of field.
Quod erat demonstrandum.