Hi Painteread,
It sounds like the sponge underlay of the rug had degraded. Once this process starts, it's best to replace them to avoid damage to your floors. The breakdown of the sponge underlay can also yellow your floor due to the chemical process.
Plain old rubbing alcohol should do the trick of removing the remaining debris.
I found an interesting site called Sultan Carpet that had many helpful tips.
http://www.sultancarpet.com/living-with-carpets.php "Rugs on smooth floors can be hazardous, and all rugs will 'walk' if laid on a fitted carpet with a deep pile. Such problems can be alleviated or eliminated by the use of an underlay. There are many different types available, serving varied functions, and your local rug dealer should be able to advise you as to what is suitable. Underlay helps to cushion a rug, and can entirely stop slipping or sliding; it also lessens any wear and tear. Any floor surface that is supposedly smooth will have uneven areas, and any bumps that are continuously walked over will eventually erode the weave and the rug will need to be restored. Underlay can allay this danger and it is considerably cheaper to replace the underlay than to have the rug restored. There are many different products available on the market, some of which are effective against movement problems, but tend to leave a sticky deposit on the floor surface and the underside of the rug. This is removable with white spirit, which should not damage either a polished floor or machine-made carpet; however, it is advisable to consult manufacturers before using any substances that might be potentially harmful. "