。
She moved to somewhere warmer for the sake of her health.
Literal
She [topic-は] health [for-the-sake-of-のために] more warm place [toward-へ] moved.
健康のために is the noun version of the purposive ために — '[noun]のために' is 'for the sake of [noun],' while '[verb-dict]ために' is 'in order to [verb].' The destination uses へ rather than に, which is interchangeable in this slot but emphasizes the directional vector ('toward there'). 引っ越す specifically means 'change residence,' distinct from 動く ('move') or 移動する ('relocate'). The implied scenario — moving south or to a coastal area for warmer weather — is a recognizable trope in Japanese discussions of retirement (保養地 'health resort areas' like Atami or southern Kyushu).