Accessibility is, as always, a huge concern. Unfortunately, Media Remoting currently lacks support for transmitting text tracks and displaying them remotely with the video. So, for the short-term, we should seek to disable Media Remoting (i.e., fallback to screen capture) wherever reliable signals are present to indicate captions are in-use.
One example:
http://iandevlin.github.io/mdn/video-player-with-captions/
Click on the 'CC' button, choose English captions. Cast tab, then start playback. See captions on remote device. Go fullscreen, we switch to media remoting, but captions disappear from the remote display.
This is using WebVTT, per https://developer.mozilla.org/en-US/Apps/Fundamentals/Audio_and_video_delivery/Adding_captions_and_subtitles_to_HTML5_video.
Comment 1 by mlamouri@chromium.org
, Mar 23 2017