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.
Assumptions:
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 src_dir to 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.
[edit] 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 Then, you could possibly build a list of files that have been archived successfully, and use the --exclude-from-file=/some/where/exclude_list.txt option.
rsyncwould simplify processing and also your human interpretation of the results.