I have done a similar trip this year with my dh (no kids). Are you planning on driving or flying between cities? We hired a car and whilst we enjoyed the driving, our trips were up to 6 h of driving which is not great with small kids (although tbf we visited 6 cities in two weeks, not just the three you mention).
In addition to the weather points made by others, I would also consider safety, and how suitable your destinations are to family travel. New Orleans has huge problems with drugs and homelessness. Marijuana smoking is everywhere. The main tourist drag of Bourbon street is NOT family friendly and is full of messy drinking from mid afternoon.
Memphis has very high crime rates, we saw a lot of panhandlers, driving in the city was crazy and every second car was dented or smashed. Nashville traffic is crazy and the public transport is non existent.
Saying that, we had a fantastic holiday, and enjoyed all kinds of experiences such as a voodoo tour, live music in all three cities, the WW2 museum, Memphis Zoo, a plantation visit, a jazz brunch, microbreweries, streetcar rides, Civil Rights museum, Natchez Trace, Peabody Ducks etc.
I would wait until your children are older.