I was looking into this some years back when I was considering building a high-availability IRC bouncer that can pass the TLS IRC connections around with this.
There isn't anything out of the box that I could find, but there was some discussion/prototyping around adding an API for exporting all the necessary key material and metadata to the mbedtls API. With that it would have been "relatively" "easy" to do the TLS bits :)
There is some security concerns with exporting the ephemeral private key material over a network connection. Technically there is nothing that makes this impossible, but from a policy perspective it may be a nonstarter.
If you're OK with transferring tcp state, I don't see why you wouldn't be ok with transferring TLS state, too. You don't even need to transfer the certificate private key. I've seen some systems where the certificate private key isn't present on most edge nodes; session signing is proxied to centralized nodes and the edge nodes just do the bulk ciphered with the session keys, which necessitates sharing the session keys over a network.
That's why I was thinking more about a key renegotiation, which I know exists in SSH and existed in TLS at one point (at least I know there's a shortcut for it in openssl's s_client).
One of the ideas of tcp checkpoint restore, at least for me, is to avoid a round-trip delay when restoring the connection. Wouldn't a key renegotiation cause at least one round-trip? I'm probably being dense here...
I'd actually be really interested in a write-up on such ideas. SSH, TLS or wireguard (voluntary) 'takeover' or Checkpoint&Restore. I don't remember whether QUIC had multihoming (might help with checkpoint restore) but since most APIs are in userland and it's udp it might be faaar easier.
