LH is still one of the few airlines that are *relatively* generous with upgrades. However, if you don't have LH
status it will be tough. I had a few times that check-in staff were very friendly and wanted to give me an upgrade (even without me asking for one), but then they showed me the long list of LH
Hons who all were in front of me (I have UA
Gold).... And, as OHLHD mentioned, with a small child, I'd see it as a miracle if you got an upgrade.