If your puppy cries in the crate at night, they are usually lonely, need a potty trip, or the crate still feels new. Keep the first nights boring and close by: crate near your bed, one last calm potty trip, no play after lights-out.
Disclosure: I'm the founder of Love My Puppy.
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Free guide: crate training a puppy on Love My Puppy
Household tips only, not vet advice. If you're worried, check with your vet.