A few technical caveats to this post:
The media type matters. I’ve found that
mp4 works fine, but
mkv doesn’t, and needs to be converted. What I found works for me is the following:
brew install ffmpeg fmpeg -i original.mkv -c:v copy -c:a aac newconverted.mp4
Also, for larger files, I find that the Casting breaks if I try and play another file straight afterwards. One way I’ve gotten around this is to close the tab and re-open the file browser in another tab, and start the cast again.
 I’m running Chrome and macOS Sierra. Any other operating system or browser combination is best-guess.