How much do urban pigeons move around cities?
I'm only aware of two studies that have looked at Rock Pigeon home range size. Note that this isolated research and may not be representative of Rock Pigeons everywhere nor of other pigeon species. 

Rose and Nagel (2006) tracked 80 feral pigeons using GPS receivers and found 32% remained within 0.3 km of their roost location. Only 7.5% traveled over 2km from their roost location. Individuals of this species typically have 1-2 primary feeding locations within their home range which ranged from 3 - 151 ha in size (Rose et al. 2006). 

Rose, E., D. Haag-Wackernagel, and P. Nagel. 2006. Practical use of GPS-localization of Feral Pigeons Columba livia in the urban environment. Ibis 148(2):231-239.

Rose, E. and P. Nagel. 2006. Spatio-temporal use of the urban habitat by Feral Pigeons (Columba livia) Behavioral Ecology and Sociobiology 60(2):242-254.