I am having this problem too, but I think the problem is actually with photostream - I am not sure that it is delivering Live Photos, but rather just the still jpg.
I don't use iCloud Photo Library due to the size of my library, preferring to use the photo stream to have the last 1000 photos viewable on iPad, and my large library on my MacBook Pro.
I have a 6S, iPad Mini, and MBP running El Capitan. When I take a live photo on my 6S it shows as a live photo on my camera roll. It doesn't show up as a live photo on the photo stream album on the phone, and is also still on my iPad and Mac photo streams. The still photo is then imported into my Photos library on the mac from the photo stream. Only problem is, I want the live photo - so I plugged in my phone.
The photos were all recognised as already imported, so I expanded that bit and manually imported all the photos I had taken since getting the 6S. Then I had duplicates of them all - and both of them, in the case of live photos, were "live" a day or so later, upon opening Photos, the duplicates had disappeared, leaving just the live photos behind. I've just manually added three more live photos from this morning - the still photos have been converted into live ones, and currently I have six photos (2 x 3) in the library - but I've had duplicates disappear twice now so these might go down to 3.
I wouldn't mind having to import the "live" part of the photo like this, if they showed up under import as needing to be imported, rather than already being in the library. As it is I can't see from import whether something is live or not, so I'm having to reimport everything manually (which negates the point of photo stream).
Ideally, live photos would start working with photo stream.