I don't post videos very often on Tumblr at all- Usually I just use freeze frames from a video. However, on the occasion when I do have a video to post, I usually use an iFrame to embed it, hosting the video itself on my site. I create a dedicated html page on my server where the video will display at 100% width, with no margin or padding or anything. I then embed this page as an iframe on Tumblr, this seems to work just fine, but I haven't really checked how it is handled on mobile either.
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