I would argue it's not the difference in wealth, but the cost of living, that entices these people to move to poorer areas.
The greater difference in wealth may in fact hold them back, for fear of criminals with even less to lose than the criminals in their home country.
2. To some degree, people do migrate to poorer areas and countries, including those of retirement age, to improve their perceived quality of life which is solely due to greater difference in wealth.