Proper way to apply plumber's grease to a two-handle bathroom sink faucet?
Hardwicke,
  I opened up the file in your letter. It is the 5.1 and 5.2 cartridges that you need to lubricate. These come apart by hand and that is what you need to do. I recommend one at a time and pay close attention to put them back exactly how they were. Or the handles may operate reversed.
  I would not use lithium grease, rather I would recommend a good food grade product like super lube 3 oz tube synthetic grease sold at home depot.
  And next time consider a higher quality faucet. This would not be happening with a Moen.