The initial conquest of England driven by the ruler of the Norman state, the duke of Normandy, OTOH the conquest of the Mezzogiorno (Southern Italy) started out as mercenaries, which managed to establish states of their own. The latter were much more fortune seekers, the family, which came to rule the kingdom of Sicily, the de Hauteville (d'Altavilla) were just a baronial family back in Normandy.
The same groups, which IOTL were fortune seekers probably want to do the same. Some might go to the British Isles instead, however other parts, like the Mezzogiorno IOTL, were more wealthy and valuable than some parts of the British Isles; some might go to the Iberian Peninsula instead....