It is possible, as I had been doing this already for at least 5-6 months now though using Kodi, with AppDaemon.
When I play any video on my Kodi, or Chromecast, AppDaemon always keeps track of the url that is being played in each room. When I move to another room, I can then instruct my system either via voice (Snips) or text (Telegram) to transfer what I was watching, and it will pause in the other room, and continue where I am.
I don’t make it do it automatically, to avoid disrupting others in the house. But making it to work automatically is easy, as I also kind of keep track of where individuals are in the house via phones.
There are some limitations though, as I haven’t been able to get services like Netflex to work yet (even if playing on Chromecast), or any services on Roku (a pain working with Roku). But any stream on Kodi, I can transfer, as you can query the Kodi for its url via Hass using the kodi.call_method service.
EDIT: If I may add, YouTube live stream been a pain to work with of late also