Take bash on tour with builtin setns.
You need the bash-builtin headers.
$ make
Load it into bash either once or in your .bashrc:
$ enable -f ./setns.so setns
For example, you can take your Ubuntu's bash into an Alpine container's filesystem (as root):
# cat /etc/lsb-release
DISTRIB_ID=Ubuntu
DISTRIB_RELEASE=20.04
DISTRIB_CODENAME=focal
DISTRIB_DESCRIPTION="Ubuntu 20.04.2 LTS"
# setns /proc/<alpine-container-pid>/ns/mnt
# cat /etc/alpine-release
3.13.1
With appropriate other bash builtins (like ls and cat), you can now browse inside a container that's FROM scratch.