Just a piece of experience from past:
Many years ago I was returning from BEG to PRG through VIE on Austrian, Friday evening. After checking in, obtaining boarding pass and going through passport control to the gate, the delay was announced. In another announcement I was asked (together with 2 more unrelated names) to go to some ticketing office (no reason given/explained).
I was escorted by a gate staff there (back through passport control) and the guy there told me due to the delay I will miss my connection in VIE. Therefore he is cancelling my ticket and will give me hotel and direct flight on Saturday noon (different carrier). Without any asking for my consent, he just handed me the printout of the new itinerary.
As I absolutely needed to be home already on Friday evening, I was insisting this is not a solution for me and that I want to fly to VIE as planned.
He was trying to convince me that it is better for me to take the direct flight next day and in case that I go to VIE and miss the connection, nobody will take responsibility for the misconnection anymore - no rerouting, no hotel, no refund (?!? he literally say in this way). I said I don't care, in such case I will take bus from Vienna to Prague on my expense.
He, somewhat angry said - ok, I am changing your ticket back.
I took the flight, we got out of the plane about 5mins before scheduled departure of my connection. I checked the screen with the flights and my connection had delay announced of 20 or 25mins. I rushed to the gate, they told me they were confused and didn't know if I am coming based on what the BEG staff did with my ticket. But I got the boarding pass from them and made the flight to PRG as originally planned.
My final impression was: Either total incompetence of the BEG staff, or a failed attemt to find a "volunteer" to get them out of overbooking situation without necessity to pay for compensation.
In any case, for vast majority of scenarios (95% or more), it does not make sense to deal with the delay in the original departure point, before the misconnection happens. Being strict and refuse offered solution might help in such case.