I attempted to bring a skeleton horseman into the Nether from the overworld on my Minecraft Bedrock realm. I had him follow me and cross through the portal, but when he entered the frame he did not teleport as expected. Instead he just sat in the frame and stayed in the same dimension. 

i could not find a solution to actually have the horseman enter the Nether. I tried leashing the horse and tugging it through, pushing him into the frames, and having him walk in in his own volition. The problem persisted before and after I name tagged both the skeleton and the skeleton horse. 
 

To recreate this simply try to have any skeleton horseman enter the portal in any way. I tried to modify the sizes of the portals in both the overworld and Nether but neither attempt rectified the issue.