The Raspberry Pi will be used as the target device for this document. If you have another target, e.g. some bitbake concoction, then you will have to adapt the instructions accordingly.
Cross compilation is described in two ways - one that uses the cross tool and one that is manual. Depending on your needs you may decide on one or the other.
The cross tool attempts to make it as simple as possible to cross-compile software by automatically fetching the appropriate cross compile toolchain and environment.
Install docker if you have not already.
$ sudo apt install docker.io
Install cross for Rust.
$ cargo install cross
Install the tool according its own instructions. Ensure your docker permissions are set. Now you can use cross in place of cargo. e.g.
$ cross build --all --target armv7-unknown-linux-gnueabihf
The additional argument --target armv7-unknown-linux-gnueabihf tells cross to set up a build environment before invoking cargo.
The cross tool may have an issue running cargo on Fedora / Red Hat dists due to a SELinux policy. Read the bug for a workaround.