Until proper Sway backport is made, here is how to backport it yourself using only Docker. Save following in `sway.dockerfile`:
FROM debian:bookworm
RUN echo "deb-src http://deb.debian.org/debian/ testing main" >> /etc/apt/sources.list && apt update && apt upgrade -y && apt install --no-install-recommends -y packaging-dev debian-keyring devscripts equivs libdistro-info-perl wget nano ranger
RUN mkdir -p /build/wlroots && mkdir /build/sway
WORKDIR /build/wlroots
RUN apt source wlroots/testing
WORKDIR /build/wlroots/wlroots-0.16.2
RUN yes | mk-build-deps --install --remove
RUN EMAIL=sway@debian.org dch -b -v 0.16.2-3~bpo12+1 "bookworm backport"
RUN dpkg-buildpackage -us -uc
RUN apt install ../libwlroots-dev_*_amd64.deb ../libwlroots11_*_amd64.deb
WORKDIR /build/sway
RUN apt source sway/testing
WORKDIR /build/sway/sway-1.8.1
RUN yes | mk-build-deps --install --remove
RUN EMAIL=sway@debian.org dch -b -v 1.8.1-2~bpo12+1 "bookworm backport"
RUN dpkg-buildpackage -us -uc
RUN mkdir /build/deb
RUN mv /build/wlroots/*.deb /build/deb && mv /build/sway/*.deb /build/deb
CMD ["/usr/bin/bash"]
Build docker image with `docker build -t sway -f sway.dockerfile .`, then run image to copy built debs `docker cp $(docker run -d sway:latest):/build/deb .` and finally install debs with `sudo apt install ./deb/sway_1.8.1-2~bpo12+1_amd64.deb ./deb/libwlroots11_0.16.2-3~bpo12+1_amd64.deb`