The Henry Big Boy is mostly just a reverse-engineered Marlin 336 with a different claw extractor, which should make little difference.
Marlins will generally pop empties out at least a couple of feet before dropping, regardless of how slowly that you operate the lever.
Both rifles DO have a spring-loaded ejector, although they are not coil sprung. Instead the ejector is held in place in a groove in the bolt and by a peg on the ejector and a hole in the receiver. The ejector is sprung outwards with a small flat leaf spring towards the ejection port and that spring assists with popping the empties out when the lever is operated slowly. In all other respects it behaves just like a stationary ejector. Like I say, Henry Big Boys are very similar.
In my experience with my Marlin 336, .30-30 cases are popped out more energetically than with the .44 magnum or .45 Colt cases in my 1894 Marlins, when the lever is operated slowly. That could simply be because they are longer and "flip" better. However, I feel that the ejection is quite good for the pistol cases.
If your ejection seems lacking, you might try removing the extractor and try bending the spring a bit to give it more tension. It might help.
As you probably know, that is just a matter of removing the lever screw and lever, and then sliding the bolt out to the rear. The ejector will just drop out if you tilt the rifle to the side.