How can I slow down rsync?

Unlike DopeGhoti's experience, the --bwlimit flag does limit data transfer, with my rsync (v3.1.2).

test:

$ dd if=/dev/urandom bs=1M count=10 of=data
10+0 records in
10+0 records out
10485760 bytes (10 MB, 10 MiB) copied, 0.0871822 s, 120 MB/s

$ du -h data
10M     data

$ time rsync -q data fast
0.065 seconds

$ time rsync -q --bwlimit=1M data slow
10.004 seconds

(note: my time output looks different to most time invocations (zsh feature), those times weren't edited by me)

Else, perhaps something along the lines of a double -exec in find. I believe that rsync -R should create & copy the parent folders, but if it doesn't, then cp --parents should.

$ find /failing/usb -exec rsync -R {} /somewhere/safe/ \; -exec sleep 1 \;

Note: also check out ddrescue, it might be right what you're looking for :)

A bit of a MacGyver solution but I have had good success with it in the past:
Put an old USB 1 hub between the stick and the computer. No way in hell it is going to copy fast that way :-)

Other possibility: If you've got another stick available put it on another USB port connected to the same root-hub (so it shares bandwidth with your problem stick).
Now start a big write operation to the extra stick so both sticks are competing for bandwidth. For further control you can use ionice to lower the priority on the problem stick even further.

You can use rsync --bwlimit=RATE to throttle the file transfer speed which, based on comments below, actually specifically seems to work by throttling the read speed of the data, which is precisely what you're looking for.