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 - Embedded Videos - Social Plugins - Documentation - Meta for Developers
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.
<html>
<head>
<title>Your Website Title</title>
</head>
<body>
<!-- Load Facebook SDK for JavaScript -->
<div id="fb-root"></div>
<script async defer src="https://connect.facebook.net/en_US/sdk.js#xfbml=1&version=v3.2"></script>
<!-- Your embedded video player code -->
<div class="fb-video" data-href="https://www.facebook.com/TheBuskersConsett/videos/253882769188513//" data-width="2000" data-show-text="false">
<div class="fb-xfbml-parse-ignore">
<blockquote cite="https://www.facebook.com/facebook/videos/10153231379946729/">
<a href="https://www.facebook.com/facebook/videos/10153231379946729/">How to Share With Just Friends</a>
<p>How to share with just friends.</p>
Posted by <a href="https://www.facebook.com/facebook/">Facebook</a> on Friday, December 5, 2014
</blockquote>
</div>
</div>
</body>
</html>