In our mei tai I can until all the straps and lean forward over a bed and lay DD down with the mei tai under her. Then, if needed, I can scoop her up and put her in the cradle. I've not had the same luck with the SSC yet. And in the ring sling I could do it with help -- loosen the rails and lift the sling off while holding DD to my chest and then put her down. I think it probably depends on your LO though.
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DD doesn't always wake up when I take her out of the Moby, but that may be because she's still in the very sleepy stage and I wear her in the newborn hug hold... when she's older, I'm pretty sure she'd wake up with me trying to wriggle her out of this thing.
Re: Is there a way to get baby out of the wrap carrier
The O'Baby Blog
This. Its called the kangaroo hold and it is specifically recommended for babies that wake up when moved.