I would imagine the majority of cases such as this very sad event mean that the police are trying to control dangerous dogs within an enclosed space eg garden or house. As such very powerful weapons are probably not needed and indeed from my thankfully limited knowledge of firearms probably more dangerous due to ricochet. If there is no time pressure then drugging dogs via food etc is probably viable (dosages can vary but at all stages the human lives would take priority so overdose would be risked)
Dart gun licences are not that common and having used them in the past to dart highland cattle the drugs are pretty scary to be using in a Controlled environment without doing it with a dangerous or attacking animal.
Trying to shoot at a rapidly moving animal eg dog in an open space is highly unlikely due to the risk of collateral damage and I imagine that police cars might well be used to run animals over if needed.
Most incidents will involve dogs rather than larger animals such as bulls etc as most cases of escaped cattle etc don’t pose a direct risk to human life once trains and motorways etc have been stopped.
