It varies quite a bit so I can only speak to my own experience. I've worked in both banks and asset managers/investors (which is my current role). For banking, some offices are almost purely administrative offices (maybe even just IT/HR effectively outsourced to a lower cost jurisdiction). Most will include at least some senior front office functions (client relationship people effectively), some are almost full setups including junior staff working on the corporate side of the house (eg a couple of banks have almost full setups in Madrid and Paris).

Very few markets front office people (ie traders) will be outside London though, which is where there has been perhaps most debate about who might move (though one issue is very few of those people are willing to move - some banks tried early on to move teams and found whole teams simply quitting to move to a competitor who didn't ask for that instead). This is where most people in the industry feel there is still some risk of further moves but the risk seems to be receding rapidly now.

In asset management, the fund itself may well be in one jurisdiction (usually Ireland or Luxembourg for reasons relating to tax transparency for ultimate investors and the flexibility of legal form offered, though sometimes Cayman Islands/Jersey/etc are used and very occasionally UK, Germany France etc). There will often need to be some form of local presence to ensure the right tax treatment there, but usually this is administrative/token and often outsourced. There might also be feeder entities in other jurisdictions again due to individual investor tax/legal requirements (fund which invests in the main fund or similar). Management for the funds can be across more than one office though this might often mirror the banking set up - bulk of junior find management staff will be in a single location (typically London) and there might then be a few senior staff in one or two other offices.

Early on the EU tried to put pressure on asset management industry to move full management of funds "onshore" but this is fraught with difficulty because it's very hard to write a rule which "penalises" London like this but doesn't have massive negative consequences for institutions who invest with funds managed outside both the EU and UK. I think this has probably gone away now.

Elsewhere, I believe some rating agency staff have been moved from London to the EU and there is an ongoing debate about clearing house location for some trades (at the moment that's staying in London because there isn't the capacity to do it anywhere else in the EU and I suspect the pressure to move will subside even more now).