Cute babies. I keep diamond doves too.
They will keep breeding and nesting no matter what you do. They will even resort to nesting on the aviary / cage floor if no nesting basket or seed dishes are there!!!
However, it's not good for the hen to keep laying eggs and raising the babies over and over again, as it depletes her body of calcium and other nutrients and leads her to get sick and even die.
I don't let mine breed at all now unless I have someone wanting to buy some off me. The other reason is each pair need their own cage otherwise they will fight and pluck each other feathers out. You will have to get a separate cage for your to babies once they reach sexual maturity I am afraid. (They can kept in a flock only in a large aviary where they have space to set up their own territories).
To stop yours breeding all you have to do is find a couple of small white balls, stones, anything round and the size of the doves eggs.
Let the pair nest in the nesting box / basket and then once she has laid her eggs throw them away and replace them with the fake eggs.
They will happily brood them for the normal time, and then give up and start again.
Never just throw away the eggs and not replace with fake ones... because that will make the female keep laying more eggs each time a few days after. This will make her ill.
I don't know what you feed your doves but I found the following is a balanced diet for mine:
Mixed finch seed
Millet Spray
Seeding Wild grasses (just pick different kinds but be careful that they are free from chemicals and not from near a road). Can be ripe or green or dry... they like variety.
Canary egg food (Its yellow high protein crumbs... they like to eat this dry from a separate dish). I feed then when they are raising young or moulting as the extra protein helps them.
Bird minerals - Oyster Shell grit and canary / finch grit.
Crushed up Charcoal (this is usually in the cage bird grit anyway, but you can buy it separate if not. This helps them with digestion.
Mine also like taking a shower with a plant mist sprayer, or outside in their cage in a summer light rain shower. This keeps their feathers soft and neat. Be careful no cats are about... keep a close eye on them when they are outside.
They will also benefit from unfiltered UV light from the sun. Just take the cage outside on a nice day (make sure they have shade if needed).
Some people say they will eat finely chopped up vegetables, but I have never got mine to eat them.
When they are moulting feathers or raising young you can add a bird multi vitamin to the drinking water, but this is not really needed if you feed all the above things and I have not used vitamins for any of my birds for years as they get a varied diet.
Sorry for long post, hope some info will help you.
Enjoy you doves!!!