You can copy the necessary terminfo descriptions to remote hosts.
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On the local system, dump the description to text format:
infocmp xterm-256color > xterm-256color.ti
infocmp screen-256color > screen-256color.ti
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Copy to the remote host and compile:
tic xterm-256color.ti
tic screen-256color.ti
The descriptions will be stored in ~/.terminfo.
(infocmp and tic are part of the ncurses or ncurses-bin packages in most Linux distributions, as well as most BSDs that use ncurses.)
send-ssh-term
can do this via one command.