Say that you deploy a new WP site with Docker (a contrived example):

docker run -d --name=mysite_com -v /data/mysite_com:/var/sites/mysite_com wordpress

A few months later a new container is released for WordPress. This new container includes a new version of PHP.

docker pull wordpress

Now you can rename your existing container:

docker rename mysite_com mysite_com_old

Stop the old container:

docker stop mysite_com_old

Start a new container, with the new image that you pulled:

docker run -d --name=mysite_com -v /data/mysite_com:/var/sites/mysite_com wordpress

If the new container is not working, stop the new site:

docker stop mysite_com

Rename it to _notworking:

docker rename mysite_com mysite_com_notworking

Rename the old container back:

docker rename mysite_com_old mysite_com

Restart the old container:

docker start mysite_com

Remove the nonworking container:

docker rm mysite_com_notworking