Great tip from @benaryorg:

A great opportunity to bring up the little known tail idiom: tail -fn +1 file

With that you can effectively output an entire file while still tailing it. That means you can stream without any complicated pipe setup:

# record

asciinema rec file.cast
# play
tail -fn +1 file.cast | asciinema play -
# play via ssh
ssh example.com tail -fn +1 file.cast | asciinema play -

Edit: for lots of output (e.g. coloured stuff like syntax highlighting in shell or editor) I recommend putting a buffer in between to account for network latency and such:

ssh example.com tail -fn +1 file.cast | pv -qB 8m | asciinema play -

Edit2: should be all uppercase -B in the pv invocations

Edit3: and there's the fitting cast: https://asciinema.org/a/163778

Edit4: the ssh-streaming part should also be applicable