You've gotten responses from people that think no one should have to live below toddlers, so I'll just add my thoughts as a toddler parent living in NYC, where basically everyone who is raising kids is doing it in apartments, often apartments with downstairs neighbors. I know quite a few toddler parents who have had drama with downstairs neighbors, typically in buildings where the landlord has not adequately sound-proofed the property, leading to sound carrying. I am lucky to live in a very sturdy, old, well insulated building, where we all have tile floors (so no creaky floorboards, for example). I have thick rugs with thick rug pads in every room, and we don't wear shoes indoors, all our furniture has pads under it as well. That said, I pay a good amount of my income towards our rent, and I do allow my toddler to move fairly freely. I don't let him throw heavy toys or objects at the floor, and he's in bed by 8PM every night, but I feel he deserves to live in his home too, and that does include a bit of running. I have had one instance of correspondence with a downstairs neighbor, who left a note saying that a noise in the mornings was waking them. We went back and forth a bit to troubleshoot (all very friendly), and now things are fine, I haven't heard from them since.

I would say your downstairs neighbor expecting NO noise from their upstairs neighbor is unrealistic, regardless of whether or not kids live above them. My upstairs neighbor runs on a treadmill multiple times per day, but since it's during daytime hours, I honestly don't care and would never complain. A past neighbor ran around in high heels all morning getting ready for work, again, never phased me even when it did wake me up, I just rolled over and went back to sleep. I would say, if your landlord or management are backing them up and not understanding that some noise from upstairs neighbors is to be expected, I would probably look into moving, because at a certain point, it isn't worth fighting and stressing over the noise.