Could anyone point to some references examining Bell inequality violations at large distances please?
I see many times, in pop science articles and research literature alike, that the quantum information of the entangled state is transmitted instantaneously to all components of the state. Strictly speaking, we must say that this is a theoretical prediction and provide an lower bound on the speed of information transfer from an experiment, right?
I think there is a conceptual thing going on here that needs clarifying (I'll leave the experimental links to others). I presume the question is predicated on the idea that, well, measurements are made within a certain time of each other, which is compared to the distance between the places where the measurements are being made. The concern is that this only gives a bound: if information is transmitted, it happens faster than some velocity which we have now bounded.
However, what one ought to do is consider what special relativity tells you: if two events are space-like separated, there is no notion of temporal ordering. Different observers, travelling at different velocities, can see the events happening in different orders (or simultaneously). So, all you need to know is that the measurement events are space-like separated (i.e. the distance between the events is larger than speed of light $\times$ time between events), and that is enough.
Also, there's a terminology issue. Bell tests do not talk about the transmission of information, but the presence of correlation. The term information would suggest that one party can choose some information to communicate to another party. This cannot happen faster than the speed of light. But the "random decision" made when a measurement is made on an entangled state is somehow resolved everywhere simultaneously, but does not communicate any information.
The largest scale Bell test done thus far is the "Cosmic Bell Test" of 2017. It ruled out hidden variables within a distance of 600 light years from Earth.
The 16 significant Bell test experiments performed between 1972 and 2018 are listed here with references to the original papers.
As far as I know the distance record was obtained in the Canary islands experiment, where Alice and Bob were 144 km apart.
There exists a paper that directly does what you want, lowerbound the "speed of quantum information". They obtain the bound $1.5 \times 10^4 c$. I read it as a work of comedy. Nobody thinks that quantum information is actually being transmitted faster-than-light in order to produce the experimental results. The collapse of the wavefunction is not physical.
There's also a subtler argument that shows we can't get away with any finite speed. If you think collapse is real it must happen literally instantaneously in the whole universe at once.
