Keep ammunition and fired cases away from anything that touches your food. Being able to clean brass easily isn't worth the risk you are placing yourself and anyone else who eats off dishes cleaned in the same appliance. If you have an old dish washer that you no longer use for anything that touches food then go ahead.
If you want to clean your brass, put it into a seal-able container with hot water and a teaspoon of dishwasher detergent. Shake it around to agitate and then rinse with hot water. Use only a small amount of dishwhasher cleaner as its very caustic.
Frankly there are better ways to clean brass. Dishwasher detergent is formulated to remove organic particles not heavy metals, oxidation and carbon scouring. A cleaner designed for industrial metal parts washers would be more appropriate. The problem is finding small quantities for home use.