Read any good geotechnical text. In general the bearing capacity increases with depth due to the surrounding overburden increase and changes in lateral stress distribution.
Are you referring to a foundation for a structure without a basement, as compared to one with a basement? and the foundation the same depth below final ground elev?
If so, at least with respect to settlement, the unloading of earth pressure by basement excavation can be replaced by a foundation pressure, in part, with little if any settlement effect.
In any case, most cases of "allowable" bearing capacity deal with "allowable" settlement, rather than bearing capacity as most texts explain it, tied to soil rupture. For the case of deeper footings, the ratio of increased soil pressure to existing soil pressure also affects resulting settlement. For the deeper you go, the less the settlement with same footing pressure, usually. There are methods out there dealing with how to compute this.
Bearing capacity is a function of the soil properties, the footing size and shape, and the overburden pressure around the foundation. If excavation near an existing foundation reduces the overburden pressure, the bearing capacity could be reduced and there could be some associated settlement.
