As far as I understand DoF is dependent on how far away you are from the photographed obect, and because with different sensor sizes you have to choose a different distance between you and the object to achieve the same framing sensor size does impact how shallow your depth of field will be. Now, If I would mount a 10mm lens set at f/4 on an APS-C camera and a 16mm lens also set at f/4 on a full frame camera standing at some distance away from my subject, I would get the same field of view and the same DoF. Is that correct?
No. Assuming everything else to be perfectly equal (which it won't be in the real world, but for a bit, assuming so), and assuming that you are enlarging to print or view at the same size, the DoF from the APS-C camera with the same framing will be roughly equivalent to that from the full frame camera stopped down by the crop factor.
That is, on a 1.5× APS-C camera (everyone but Canon), you'll get roughly equivalent results stopping down the full frame camera to f/6; on 1.6× Canon, f/6.4. Of course — real world being pesky again — you may not have those stops on your camera; f/6.4 is a third-stop down from f/5.6, so that's likely, but f/6 is only about a sixth of a stop down, and few lenses/cameras give you that fine of control.
Both lash-ups yield the same angle of view. 16mm mounted on a full frame vs. 10mm mounted on an APS with 1.6 crop factor is: 73.7⁰ height by 96.7⁰ length by 107⁰ diagonal (diagonal most frequently published).
The depth of field is not the same: Depth of field has more ingredients than you listed.
Focal length of the lens
Distance focused upon
Degree of magnification applied to obtain viewed image
Distance viewed image to observer
Acceptable size of circle of confusion
The span of depth of field is a subjective call. This involves the acuity of the eyesight of the observer, the contrast of the image, the distance observer to image and several other factors.
A frame with a crop factor of 1.6 means that it is 1/1.6 = 0.625 X 100 = 62.5% of the size of the full frame. The difference is: Full frame = 24mm height by 36mm length by 43.27mm diagonal measure vs. 15mm height by 22.5mm length by 27mm diagonal measure.
The key here is: To make the same size displayed image, the smaller frame size must be enlarged 1.6X time more than the full frame. In other words to make a 8 X 12 size print or computer displayed image, the full frame must be enlarged 8.5X vs. the compact must be enlarged 13.6X. We compute depth of field based on an acceptable diameter of the circle of confusion as projected by the lens. Since the compact frame size requires more magnification, its depth of field calculations are more stringent. Anyway, all depth of field calculations are subjective.