You are right to wonder about the effects of diet on urine volume at night, but I don't think that it's necessarily anything abnormal happening. Also, it's perfectly normal for 3 year-olds to wet the bed at night.
The kidneys are constantly balancing free water (solvent load) vs. osmotically active particles (solute load).
Sugars, proteins and salts (minerals) all become osmotically active after being absorbed and/or broken down by the digestive system. When you ingest osmotic particles, such as sugars, salts and amino acids, it makes your kidneys work harder, and you urinate more to restore osmotic equilibrium. This is why diabetics urinate more when their blood sugar is high--there are too many "osmoles" of particles in their blood.
If your children are healthy, gaining weight well, and are not dehydrated, I doubt that they're overwhelming their kidneys by eating too much protein, salt or sugar.
Let's look at the flipside. If you drink a lot of water, especially in the evening, your kidneys will also produce more urine to get rid of excess "free water." In severe cases, you can overwhelm your kidneys by drinking massive quantities of water, which is called "water intoxication" (a very rare condition).
You can test for urine dilution using a "specific gravity" urine dipstick, which you can get at most pharmacies. When the diaper overflows, test the urine specific gravity. If it's less than 1.010, that's considered dilute, and is usually caused by water intake. If you decrease evening water intake, it will help speed nighttime dryness.
If the urine has a large volume with a specific gravity greater than 1.010, then you can test the urine further with a "multistix" dipstick, which includes glucose (sugar) and protein. If there are any positive tests on the urine multistix, then you need to consult your child's doctor.
Again, I doubt that there's anything abnormal going on--just normal kidneys doing their job.