All the following commands assumes we saved MySQL root password into MROOTPASS variable:

export MROOTPASS=<mysql root password>

The most straight forward method to backup all the MySQL databases and all the website files (PHP scripts, images, etc..) stored in the /home directory is the following:

mysqldump --all-databases -u root -p$MROOTPASS | gzip > all-databases-$(date '+%Y-%m-%d_%H-%M-%S').sql.gz
tar -cvzf home.tar.gz /home

If we backup some individual database (probably not as root) and change its user while restoring it, it might make a sense to remove DEFINER from the output script:

sed -e 's/*]*\*/\*/'

The following commands restore all the websites from the archives:

gunzip -c all-databases-2017-05-23_15-31-00.sql.gz | mysql -u root -p$MROOTPASS
cd /
sudo tar -xvzf home.tar.gz

After migration from MySQL version  14.14 Distrib 5.5.54 to 14.14 Distrib 5.7.18 (I do not know what is the difference between them) I got the following error: “ERROR 1805 (HY000) at line 1: Column count of mysql.user is wrong. Expected 45, found 42. The table is probably corrupted” while trying to drop some user, and fixed it by running:

mysql_upgrade -u root -p$MROOTPASS
service mysql restart