From what I've seen looking at a large number of old shooting pictures from the late 1800's, the Marlin Ballard is the commonly seen rifle used.
The Stevens began to be more popular in the late 1800's, and more so in the early 1900's. But they didn't have the 44 or 44 1/2 framed target rifles earlier, so that likely explains the change.
The Winchester didn't even arrive on the market until 1885, and is a late comer also. Five years later in 1890 the Ballard ended production, so both Winchester and Stevens became more prevalent in matches towards the turn of the century. But Ballard rifles still were plentiful as used guns, and many were modified for various styles of shooting.
In long range matches the favorites seemed to be Sharps and Remington rifles. They had a history of strength, and large caliber chamberings. The long range Creedmoor type matches were losing interest in the late 1800's, but those guns remained effective, and popular.

The custom makers reflected the era they worked in. Many worked in the mid to late 1800's, so the guns they chose were guns most popular in their time. Schoyen built far more Ballard rifles for schuetzen and benchrest than he did any other action. Pope being younger, and living well into the next century built far more Winchester rifles because they were more available, and newer during his time. And late in his career he built a fair amount of bolt actioned rifles because those were the gun of that time period.
If you compare custom makers guns to the era they lived, it had a big influence on what they built.

RSW, yes I'm talking about the schuetzen rifles of the common manufacturers, nothing custom.  For example, would Franklin Mann have found that one maker's rifle to be more accurate than the others if he shot them off his rest.

Old-Win wrote on Oct 15^th, 2018 at 11:37am:


He'd have to have tested several individual guns from each maker to arrive at any general conclusion, & even then, there'd be exceptions--some particular one that happened to be far more accurate than all others of the same make. 

Furthermore, if one make consistently outshot all others, that fact would quickly become known, & nobody would shoot the other makes--which never happened.

Of course Red, but what I'm I'm trying to realize is what Vall has noticed, that you see more Ballard's in the pictures than any other rifle so was it because shooters thought they were more accurate or what? Was it the time frame that they were made?   The famous picture of Pope sitting in the chair, Walunt Hill, 1935, 2 Ballards and his highwall.

Shooters win matches, actions don't - they might have some advantages when compared to others, but the influence of the action on rifle accuracy is not that big, assuming a decent action. Barrel and chambering are more important, but the main factor is 'the nut behind', especially when shooting offhand.
A tough one to 'copy'...

Re. Franklin Mann,
Having his book, I recall that he mainly worked with "gilt edged" Pope barrels.  The barrels were securely clamped/ mounted in his "machine rest" and it probably almost did not matter what action he used.  Also Mann's studies largely worked on bullets,bullet defects, etc. 
Oh , he was interesting, but the book went on and on, with
what some would find were boring details of the studies
beltfed/arnie