Can't speak about morning EMH, but have done evening EMH. I found that Epcot was pretty good after hours. I went first week of Jan., and after 11pm, I was able to walk right into Mission: Space (no wait), and the listed wait time for Soarin' was only 20 mins (FP for Soarin' had all been gone by noon that day! and lines were up to 160 mins during the daytime). I actually got into Soarin' in less than 10 mins because I was the only single rider, and apparently they needed to fill up single seats - my wait time ended up being less than 10 mins. Also, I was able to meet all the characters in Character Connection without any wait - and got a group picture with Mickey, Donald, Goofy, Pluto, and Dale!
As for MK from 12-3am, crowds are fewer but can still be significant for some rides. I was there on Jan. 1, and at 1am, Peter Pan still had a 30min wait, and Space Mt had a 20min wait. Some other rides were walk-in (it's a small world, the casousel). But I think the later in the night you get, the fewer the people there will be.
IMO, evening EMH is also a good time to meet some characters. More show up and lines can be shorter for some of them (not Mickey and Minnie, who always have long lines regardless of the time).