push back on/over something

[GoodTaste]

USAToday's headline: Police unions push back on vaccine mandates, clashing with politicians
When clicked, it shows: 'Eye-opening moment': City leaders, police departments push back over impending COVID vaccine mandates

Source: USAToday

I wonder why "on" or "over" is used - why not just "push back (the mandates)"? So I guess the difference between "push back on something" and "push back something." The former probably means "push back parts of something" while the latter "push back something entirely." I am not sure. It appears not holding water because "vaccine mandate" is something NOT reducible to its parts. You either get vaccinated, or not vaccinated.
So whatt does "on" or "over" mean there?

 

[jutfrank]

The preposition over means something like 'on the subject of'. The preposition on is used in a similar way.

We disagree on what to do.
They fought over immigration.

So in this case, vaccine mandates doesn't actually mean 'vaccine mandates', but 'the issue of' vaccine mandates, loosely speaking.