Can I ask Christy, is there a point at which they can or will sort it out themselves? If humans get involved does that slow the process down?

I only ask because I know in a 'pack' situation they would sort it out themselves, right? I understand they could hurt each other, but how is the 'top dog' position going to be decided without them working that out between them? Or (as horrible as this sounds) would they really just fight to the death over this?

Arimeh - if the beagle is the boss I would feed her first and let the boxer see her being fed first. Feeding at the same time is very diplomatic, but far too subtle for a doggy brain! It will only reinforce to the less-dominant dog that it should continue to challenge for top spot. Same as giving treats, etc. Beagle gets everything first. Otherwise they will be confused.