to a lower place vs at a lower place

to a lower place and at a lower place both are adverbs.

Word	Noun	Adjective	Verb	Adverb
to a lower place	No	No	No	Yes
at a lower place	No	No	No	Yes

As adverbs, to a lower place and at a lower place are synonyms defined as:

Other synonyms of to a lower place include below, beneath.

to a lower place (adverb)	at a lower place (adverb)
in or to a place that is lower	in or to a place that is lower