LSD keeps the wheels locked together normally through clutches in the carrier. If a tire wants to spin faster as in a turn, with a wheel in the air, or spinning on ice, it has to overpower the clutch packs first since that's whats keeping it the same speed as the opposite wheel.
Though the process is very gradual and linear so it's not like they are always slipping on and off. The idea is the "up until it slips" part doesn't last very long because the transferring of torque to the opposite wheel has already alleviated the problem that caused the other tire to want to spin faster in the first place (eg: the vehicle moved forward even with one wheel spinning because the axles were locked and now that wheel is no longer in a patch of ice/dirt/etc.).
Note that there are two cases of the wheels spinning at different speeds, but only one that is desirable: one where one wheel has traction and the other is spinning (dirt, ice, wheel in the air), and the desirable case where both wheels have traction (as in a turn). Only the second case really ever causes the clutches to slip and allow independent wheel speeds, in fact the only condition in which there is enough force to actually overpower the clutches; the only time you want it and the whole purpose of a differential in the first place. This is the limited time the differential is allowed to slip, eg: LSD.
Without a mechanism to selectively lock the wheels together (more like unlock, they are locked by default), the wheel spinning will get all the power (twice the speed even) and the wheel on asphalt will be like the wheel being held in place in the video above. Locking the wheels together solves that problem. But when both wheels have traction, but are spinning at different speeds, as when turning, the clutches are able to slip and allow the car to turn without scrubbing the tires as with a locker or a welded differential where the wheels are permanently locked.
As I said before, the differential is about coupling two different wheels that can travel at different speeds to a common drive shaft spinning at one speed. It's function is completely independent of the suspension system, which has a different and unrelated purpose of keeping the wheels in contact with the road. An IRS still has two axles (so does a solid axle), the difference is the axles can MOVE (as in up down left right anything other than SPIN) independently rather than moving together in the single suspended housing. Note that "solid axle" isn't really solid, but two axles in one housing that can only spin different speeds through the differential, but must articulate together because they are fixed to the same housing.
So I guess we can simplify the original question:
-the differential controls how the wheels SPIN/ROTATE relative to one another
-the suspension setup, IRS vs. solid axle, determines how the wheels MOVE/ARTICULATE on the suspension relative to one another.
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