I'm a fan of the palm method. It is a bit faster. I shoot a lot of CMP matches. I find I can shoot my Model 1903a3 as fast as my Garand. Not meaning cyclic rate of fire, but aimed shots.
But going to the Krag, I've never found a smoother, faster action. My Model 1917 is a real wrestling match. I tried using the finger method, but I ran often run out of time, even cross fired in attempts to get the shots off.
I then went back and re-studied the writings of CPT. Edward Crossman on the subject. Slowed down and worked my way up to the timing I need to get the shots off in time and accurately. My last match with the M1917 was a heck of a lot better, time wise and was the best match I fired with this rifle.
With my Winchesters I wouldn't consider any other method. I have a 308 M-70 Target Rifle built by the AMU. It rivals my Krag.
As to the Mosin, In my CMP GSM clinics I teach to use the palm method with it. Palm up, pushing the bolt up, rolling back and pushing the bolt back into battery and its fairly smooth.
The trick as told by Crossman is not operating the bolt by the number, but one smooth act, palm up, rolling and slapping.
I thought about making a video on working the action using bolt with the palm method on different rifles but I haven't had good luck with posting videos as they seem to draw a lot of unrelated BS that gets the topic shut down.
One thing Crossman points out, in the rapid fire stage there is no reason the stock of the rifle should never leave the shoulder, except, of course when you load your second 5 round clip.
If one is really interested, I highly recommend reading Crossman's "Military and Sporting Rifle Shooting" on the subject.
I started HP shooting in the days when it was difficult to get a CMP Garand, (mid 70s) and M1As were just popping up, where many civilian shooters used bolt guns, Springfields and Winchesters. It was a pleasure to watch some of these guys shoot. There was very little movement of the shooter during rapid fire. Now, it seems to be a lost art. Watching some shooters shooting rapid fire with a bolt gun looks like somebody laid down on a pile of fire ants.
Dry firing and a bit of practice is all it takes. The Springfield and others its no problem, but on the M917 and similar rifles, you need to lock the follower down to keep it from locking the bolt. I found a nickel works well, sliding it under the rails, holding the follower down so it doesn't lock the bolt to the rear.