I think that if you are going to have the campers divided by age group, you're going to have to make some modifications to make it fair - like if you wanted to do a baseball game, fine, but the older group has to bat or throw with their non-dominant hand or something similar. You could do a basketball shoot-out with different distances for different age groups. If you wanted to do flag football, give the younger kids one shorter flag each, and the older kids several longer ones.
The first camp that I worked at had staff events during color war, and the campers loved watching them (and got points for cheering for their staff members.) You could try to do the events that would be more uneven with staff - though the campers are different ages, the staff members from team to team would presumably be more evenly matched.
Something like a cardboard boat competition could be pretty even. Yes, the older kids might have better engineering ideas, but they also are going to have to float a larger person in their boat than the younger ones.
Another idea would be to not necessarily avoid points based on ranking the groups, but on how well they do individually. Suppose that one of your activities is to make a team banner. Instead of giving an 1st, 2nd, and 3rd place and assigning points based on that, you could tell the judges that the activity is worth up to 10 points, and they can give any amount of points to each of the groups - so all groups could potentially get ten points.
Relay races with a variety of activities work well too. My brain is a little too fried right now to come up with a ton of ideas, but you can pretty easily create a mix of activities where different ages of campers have advantages. An older kid might beat the younger kids in a straight footrace, but the younger kids might be better at a wheelbarrow race, for example - if you put a mix of different events in the relay, you can even it out.
