To move to a new place.
is simply to change location from one place to another.
To move over to a new place.
"over" is usually used when a contrast is emphasized
We moved over from the west side to the east side of town.
We just moved over from the UK.
To move in a new place.
She moved in her new studio.
she danced in her new studio
means to make movement within an area.
To move in to a new place.
We moved in to a new home.
To move into a new place.
Let's move into the house, it's about to rain.
is to change location and move inside an enclosed space.
"Move in" also has the meaning "to get closer".
We moved in to have a better look.