A transfer of consciousness? I don't get you.
TOS MU is the only good MU. Much of DS9's was awkwardly awful and dull.
Their "souls" crossed over into their alternates bodies.
Guy gets it. Though if I'm on his wavelength, maybe I should worry.

In "Mirror, Mirror", Kirk's landing party gets scrambled in the transporter beam during an ion storm and their minds are intact, but they're in MU clothing.
With DS9, the first crossover happens during travel through the wormhole that goes awry, and we wind up with two Kiras in the MU because she and Bashir arrived in their runabout. There wasn't any changing of places with their other selves, or we should have heard about them being captured during the resolution. Later trips were done via transporter, but still seemed without bumping their doubles into the opposite universe.