I completely know what you are talking about in Rome, I saw this all of the time. They prey on tourists there.
The scenario there usually was 2 women with babies in a sling around their necks. If you are heading to the train station with luggage, that is when they hit you. They come up flock to you, when you are kind of helpless and ask for money. Its not just that they ask, they get up very close to you with their babies and harass you. It was crazy. I personally never got bothered, but it happened to my friend who was with me twice. Watch out!!! Fortunately, they never managed to take anything, but the experience, definitely shakes you up a bit.
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