I suspect it depends on your family background. I am of (Irish) immigrant stock and, even in the first half of the 20th century, both sets of grandparents moved house several times as they moved from labouring to middle class.
But I think moving really got going in the 1960s when first time buyers, unlike their parents, could no longer afford to buy a family sized house . A housing shortage was driving house prices up so you bought a property, however small, to get on the housing ladder, and then upgraded as your household outgrew its accommodation.
Increased mobility of households, increasing longevity and more women returning to work meant that the informal care networks of non-employed women caring for elderly relatives in their own homes or having them live with them broke down and the need to look after an elderly relative with increased infirmity, who was living a decade longer than a previous generation meant that there was a social need for alternative accommodation for older people, sheltered flats, care homes, and nursing homes for the elderly.
It has been a short step from that to the beginning of a sentiment that there is a moral imperative on older people to downsize when their children leave home or when they retire.