Yes, you can use both to and into in that way. Generally I think there is a subtle difference between moving to a new house and moving into a new house. Moving into a place refers to the final part of the action, when you're moving your stuff into the new house and setting it all up, whereas moving to a new place refers to the whole process of moving out of your existing house, transporting your stuff, and moving it into the new house at the other end.
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KrisBlueNZYes, you can use both to and into in that way.Generally I think there is a subtle difference between moving to a new house and moving into a new house. Moving into a place refers to the final part of the action, when you're moving your stuff into the new house and setting it all up, whereas moving to a new place refers to the whole process of moving out of your