We investigate the longitudinal and transverse momentum distributions of charged fragments originating from reactions with radioactive, neutron-rich beams. It is shown that the width of the narrow peak of the longitudinal momentum distribution is insensitive to the details of the collision and the size of the target nucleus. In contrast, the width of the peripheral region from which transversely moving particles originate is significantly narrowed via absorption of the outgoing neutrons. This diffractively broadens the width of their transverse momentum distribution, in a manner which depends on details of the collision, and so makes the transverse distribution less reliable than the longitudinal one for measuring the size of the original neutron halo.