rsync has a very comprehensive manual page, there's even some examples which indicate how to do this. Here's my take along the lines of what you want.
rsync -auv remotemachine:/path/to/srcdir/ /path/to/dest_dir
I have made some assumptions, update your question with additional information if your setup differs and you need more help.
remotemachine has rsync server set up. (ssh access is possible but then the syntax is slightly different)
-a archive mode sets several options automatically, but it is usually what you want for backups
-u update mode, only copy files that don't exist or size or last modification time differ (source files are newer than what you have already)
srcdir/ The trailing slash means copy files within
dest_dir to avoid an extra level in destination.
-v verbose mode so you can see what is happening.
To Test this, use the
-n option in addition, this skips the copy but shows you what it would do.
But I do recommend, reading the manual page
man rsync and testing.
 After @Dev 's 1st comment below.
If you have an intermediate step, ie dest_dir is really a temporary local storage location, used for offline archiving. Then, you could possibly build a list of files that have been archived successfully, and use the