I would suggest that these were probably designed to run dry or very lightly oiled (ie: no grease) and that some previous owner took it upon them self to "fix it" using the wrong grease. There will most certainly not be such a thing as a grease specific to Morse keys. (So a quick internet search for "precision bearing lubricant" will give you more info than this site).
I would first clean them thoroughly with a solvent to remove all old contaminants (even WD40 if you have nothing else). Then use a very light oil primarily for rust protection (assuming the bearings themselves are not brass) and should not use "grease".
You really don't want any noticeable lubricant build up. You may even decide to oil it well but then dry of the components with a shop cloth (which doesn't leave any fibers behind, unlike a paper towel). This would leave only a microscopic film of oil filling small surface irregularities within the cups/bearings.
Grease isn't really necessary when these bearings aren't actually seeing any serious mechanical load (rotation/torque), there is no need to facilitate heat transfer, etc. Grease is intended for application with serious work being done such as machinery/automobiles, etc and facilitates repeated rotation and torque loads, heat transfer, noise control, and rust prevention. A Morse key in your ham shack should not require any of this.
As you are an engineer, I would expect some precision in your wording:
The grease in the cups has long since turned into some sort of abrasive
Grease will not "turn into" an abrasive (as in it won't naturally degrade into an abrasive on its own). It may collect abrasives (ie: dirt/sand/etc), but this does require significant exposure to undesirable contaminants.
Old grease/oil will easily clean out with a solvent, however "abrasives" may be more stubborn. You may need to try using a syringe with a needle to create a little precision pressure washing with your solvent to displace contaminants caked in the corners. You may also need a precision scraper if you have dried/caked contaminants in the bearing cups.