Go out onto North/South Boulevards for another Holo Clip from Looker, saying that he has another task for you. Back to the bureau we go! He's worried about how Emma's been running off lately: Looker believes the Lumiose Gang to basically be on the straight-'n'-narrow, but there's always trouble brewing in a metropolis like this.
You'll soon get a bulletin, interuptting the conversation, notifying you that there's been vandalism in the Lumiose Art Museum, the destruction of a painting most importantly. Head onto North Boulevard, then just a bit clockwise and across the street to the museum. The defaced painting in question is in the center of the third floor.
Speak with the director there and then leave. Just outside, you'll get another bulletin: Pokéball thefts have been occurring across the city, particularly after Pokémon battles in back alleys. Looker calls soon thereafter, requesting you return to him. He comes up with a sting operation to get the culprits, set in the back alleys of the North Boulevard.
There are three alleys back there: one around Route 13's gatehouse, one around the Galette Stand, and one near the Station. Begin with the alley across the Galette Stand and to the left (also left of the entrance to Autumnal Avenue), then battle the woman at the end.
Trainer Battle: Suspicious Woman ???
After the battle, the woman attempts to take your Pokémon, before fleeing from Looker. Repeat this process near the Route 13 gatehouse, further counterclockwise from there.
Trainer Battle: Suspicious Child ???
Next, go far along the other way to the alley across from Lumiose Station at the opposite end of North Boulevard. There, we find that someone else has just been victimized. It's likely the culprit went to South Boulevard, and there's only one alley there: continue clockwise from Lumiose Station into the next area.
Now go into the alley where you first found Emma (with the Skiddo next to it); it's only a few paces from the entrance whence you came. Once inside, defeat the person lurking at the end of the alley.
Trainer Battle: Suspicious Lady ???
Once the battle is over, you and Looker will have the Pokemon thief cornered. However, the situation reveals itself to be more complicated than Looker had imagined and this mystery will remain unsolved for a while longer...