Grease will always drop out to an extent, but if it's doing it excessively, then it's a sign you're putting too much grease in.

For a spindle, I'd only be looking to inject maybe a single grease gun stroke every 50 hours of spindle run time, and I'd be injecting it slowly with the spindle running to ensure it's evenly distributed.
Bearings don't need a lot of grease (typical bearing load is only 25-30% of available free volume), and too much grease will do damage (it can cause the balls/rollers to slide, rather than rotate).