I have noticed a problem with the Tel Mithryn magic lift as Lydia goes up first but when I get to the top myself she is not there, but appears afterwards, coming up again in the lift but has been injured. I think she is jumping down again to be next to me before I get to the top. Has anyone else had this problem and is there a way to prevent it without leaving Lydia behind?
This appears to be fixed in the latest Unofficial Dragonborn Patch 2.0.3.
This is currently a bug in skyrim, and the only way to fix it is to download a mod called Unofficial Dragonborn Patch. If you're on PC I would recommend you download all of the patches, as it fixes a lot of the common bugs! :)