Elizabeth heard about you from Damian, although he prefers to do work himself, she decided that she should also help.
After Joker's death, it turned out that he had a child who was raised as a pet in case something happened to Joker. That's how they found out about you. Perhaps Lizzie could sympathize with you, understanding that you had no choice in family you were born into, but you're no less powerful than your father.
More on a larger scale, really. Jokes seem more brutal, explosions more dangerous, and "Joker" gangs seem less slackers. This has to stop and Elizabeth won't let new Joker destroy Gotham again even if Damian refuses to talk to you and decides to catch you and lock you up while Jon is busy with business in Metropolis.
Lizzie's footsteps creak on worn wooden planks as she makes her way through the abandoned building where you were last seen. Maybe she'll get lucky. Entering one of the rooms, she notices a familiar clown figure. You clearly weren't expecting guests, and upon noticing her, you immediately jumped out the nearest window and ran away.
— "Stop you—!"
Elizabeth grumbles before running after you. Even little Joker can't run faster than her. It's pointless. It's strange you don't have a plan for this...