Noxians I believe would be initially distrusful, if not outright hostile towards outsiders and non-humans....only at first at least. Once an individual either proved themselves as strong, capable, or somehow of use to Noxus I think they would likely be accepted. That said, Noxians would NEVER accept refugee's unless they were their own citizens.
Demacia at first would seem more accepting, less open hostility and refugee's would of course be cared for openly...but for those trying to live their they would likely find MANY doors for advancement would be completely closed to them because of Demacia's caste system. Not only would they lack a family name to rise to a position of importance they would also lack natural Demacian birth
Zaun and Piltover are both likely progressive enough that all people would be welcomed, though in Zaun outsiders would likely be more at risk of the Mega-corporations penchant for abduction
Bilgewater is pirate town, anyone and everyone is welcomed unless they are trying to restrict their freedoms.
Bandle City at least would be welcoming to visitors, but im not sure how they would be about non-yordle permanent residents. Rumble proves there is a portion of the population that dislikes the big-folk
Freljord would depend on the tribe you tried to join, Sejuani is the more traditionalist, she would likely turn away newcomers, while Ashe would be more inclusive
Ionia it would depend on the reason and the individuals involved. Most parties would likely be welcomed, but encouraged to adopt their beliefs on balance. Noxian migrants however would face significant opposition.