I assume you refer to holes much smaller than 1mm diameter.
You may want to use acetone or/and ultrasonic cleaner since they're readily accessible.
If you have access to CO2 cleaner, it's worth trying it out. It uses solid carbon dioxide particles to blast the surface, and blows debris out of the hole. And it leaves no residue as CO2 particles turn into gaseous CO2.
Yes.. the holes are less than 0.3 mm diameter. Will acetone not interfere with the composition? When should this cleaning be done i.e., immediately after drilling the hole or after some time?
I don't think 'when' would be relevant. If oil/grease has already solidified, it will be dissolved in ultrasonic bath with solvents.
On a side note, I am investigating CO2 cleaning for small features so I have access to CO2 cleaner. In case you want to send me a sample, I'd be glad to try cleaning it for you.
Co2 snow can cleaning through holes and there are some special nozzles for holes not drilled through. Important factors are the hole size, nature fo contamination, etc. Testing is always wise. See www.co2clean.com