You could easily answer this by comparing crop frame DSLR and mirrorless. Don't forget that there are advantages to getting rid of the mirror flipping up causing vibration every time a shot is made, but then live view does that. Getting the back of the lens closer to the film plane is also supposed to be advantageous, but then you need good glass and as yet there aren't really any for mirrorless.
This is a kinda of in theory type question, so there is no absolute answer, because we do not know what technology may be in it.
In theory, it can be the same, there are all sorts of patents on autofocus for mirrorless cameras that purport to allow fast autofocus. The problem is that we have not seen them in action, so they may or may not be as good as claimed. And, there are always drawbacks that you get with a new technology.
The Nikon V1/J1 technology could be put in a FF body and sensor, for example and we would see many good features, but also some drawbacks that we might not like.
You could of course have something like the K-01 and merely remove the mirror on an existing FF camera while keeping the DSLR mount and flange distance.
The main probablem with smaller flange distances to me seems to be with wide lenses, from what I'v read digital sensors cannot handle light hitting them at as oblique an angle as film did which makes boarder performance a challange.
A Canon FF DSLR with live view locks the mirror up and, in effect becomes a FF mirrorless body. The newer ones use slow phase detect AF or they drop the mirror to do a quick af anf raise it again. You must use the lcd as a viewfinder.
The image is the same quality in liveview. A mirrorless body would, be the same net effect as the mirror lockup, except we hope that there is some sort of viewfinder, and fast AF. Therein lies the problem.