The process of sharing Apple live photos has mostly been restricted to native apps, up to now. The Facebook app, for example, detects a live photo and allows you to upload it as such. There are also third party apps, including one by Google, that converts live photos into GIF format.
However, I understand that Tumblr brought it to their web site not more than a year ago. I think they might have given some details on their implementation but until then it was basically something that no one had done yet.
That said, only this month, Apple have released their own "LivePhotoKit" JS API. So if an add-on developer wanted to work on something like this, use the following as a starting point: