In order to properly update Kali Linux on a USB, persistence must be setup. If persistence is not setup, re-imaging the USB with an ISO from the weekly build will be a suitable update.

The best way to update Kali on a USB is to follow the same way you would on a full install.

First ensure that etc/apt/sources.list.d/kali.sources is properly populated:

kali@kali:~$ cat /etc/apt/sources.list.d/kali.sources
# See https://www.kali.org/docs/general-use/kali-apt-sources/
Types: deb
URIs: http://http.kali.org/kali/
Suites: kali-rolling
Components: main contrib non-free non-free-firmware
Signed-By: /usr/share/keyrings/kali-archive-keyring.gpg

kali@kali:~$

After that we can run the following commands which will upgrade us to the latest Kali version:

kali@kali:~$ sudo apt update
kali@kali:~$
kali@kali:~$ sudo apt full-upgrade -y
kali@kali:~$

Using this, the kernel will not be updated however. If the kernel is needed to be upgraded, perhaps there is a security vulnerability patched, a re-write would be needed with the latest ISO. If there is data that is needed to be saved, an rsync to a separate storage device would be a good way to backup that data.