[SOLVED - PEBCAK] Multiple php versions vs. changing php version of existing site

Hi,
I have several php version installed on debian 8 + ispconfig3.
Technically theye are working, I can change the php version of a site and the system put its config file into the correct place and able to start with the required php version.
But.
I found that, if I change a php version for an existing site (say from 7.1.4 -> 7.0.18), the config file moved as expected, but the php-fpm processes did not restart to make it work.
Instead I have to restart the old php-fpm and restart the new php-fpm to get it work, as the example above:
I restart php-fpm-7.1.4 (it will remove the changed website from the running process) and I restart the 7.0.18 -> which will pick up the new config and serves the website.
So, my question: how does it suppose to work? What steps completed after one change the php version of a site?
These steps are completed (old restart, new restart) by cron once a day or similar?

 

Thanks.
I checked, I have /etc/init.d/ files and /etc/systemd/ directories, too, populated by a lot of services, including these php-fpm files.
As I did not really messed with init.d/systemd before (I did not really catch the necessary of systemd at all, but this is an other story), so, I am not really sure about how to left the system do its job.
But it is true, when I tried to stop/start services, I used the formula: /etc/init.d/php5-fpm start/stop
In the systemd era it seems a bad habit, need to change.
I restart the server to get a clean view, because at this moment it is sure, it messed up by init.d scripts.

 

Confirmed: if I do not play with /etc/init.d/php* manually, systemd is the only service which handling the system and can restart services in correct order which is necessary on php version changes.
Thanks!
Anyway, I did not find clear answer for this question in the manual (pdf), I only assumed it should work. Of course, it did, until I issued init.d script. PEBCAK again