Looking for a word (or set of words) to describe someone who moves their residence continuously over the years?

As a noun, nomad.

As an adjective, nomadic.

That said, there are many synonyms or metaphors you could use: unsettled, rootless, peripatetic...

As both adjective and noun, itinerant:

Adjective: Travelling from place to place.

Noun: A person who travels from place to place.

At the risk of offending the politically correct, I would also add gypsy:

A nomadic or free-spirited person. ‘why should she choose to wander the world with a penniless gypsy like me?’

Peripatetic is a word for someone always moving around, not living in one fixed place. "Perpetual tourist" is a phrase in use (see Wikipedia) for the class of wealthy peripatetic people avoiding becoming tax resident anywhere.

A frequently used idiomatic expression (often as an exaggeration) for this situation is: living out of a suitcase.
See the example sentence below from ODO.

"I'm living out of a suitcase; there's no point spending that much money (on a relatively immovable asset)".

ODO:

live out of a suitcase
PHRASE

Live or stay somewhere on a temporary basis and with only a limited selection of one's belongings.

‘Her parents told her she had to think about buying a home instead of living out of a suitcase.’