On purpose slow copy support?

[jb at dslreports.com]

I'd like to use rsync on a production box that has a great deal of
memory-cached filesystem in use at all times. Is there any way, or any
other tool, that will do a "slow" copy? purposefully not busting the
cache by copying things as fast as possible?

"nice" does not work, of course, as that is a cpu scheduling control.
What I want is basically lowest-priority or "background" I/O.

thanks