I don’t think anyone has made a Facebook Live embed plug in yet. A few people have suggested that Facebook wouldn’t allow it as they want you to stay on their site, but as this link demonstrates, that’s not the case at present - https://developers.facebook.com/docs/plugins/embedded-video-player/
I’ve managed to embed one of my ‘live’ videos on www.thebuskers.co.uk with the following code. The default video is a width of 500, but I changed it to 2000 to get it to fit the screen. It seems to shrink down okay, but it doesn’t seem to behave responsively. Is there a way to alter the code to get it to be responsive?
Code example below.