Seen a few reviews of this film so far, and nobody seems to mention Tokyo Godfathers (2003), another 'found family' tragicomedy about bringing up a baby.
Synopsis: Sang-hyun is always struggling from debt, and Dong-soo works at a baby box facility. On a rainy night, they steal the baby Woo-sung, who was left in the baby box, to sell him at a good price. Meanwhile, detectives were watching, and they quietly track them down to capture the crucial evidence.