The Austrians had upgraded to the M1895 Mannlicher mostly I think, plus the M1890 Mannlicher carbine.
United States has Springfield trapdoors out the door. There's also the Remington-Keene rifle which was used by the Bureau of Indian Affairs police. And the Army had bought 10,000 Winchester 1895s during the Spanish American War but hadn't sold them off until 1906.
Russians still had large stocks of their Model 1868 and Model 1870 Berdan rifles.
French had their Fusil Gras mle 1874, the Gras rifle.
Italy had fairly recently switched over to the M1870 Vetterli and M1870/87 Vetterli.
Germans had a half-dozen variants, modernizations, and upgrades of the Mauser Model 1874. There's also the Gewehr 88 and its carbine plus modernizations which were common through to WWI in reserve units.
Not a whole lot of them in the first-line national armies, but available in a thousand calibers and variants is the Remington rolling block. There's the accompanying
Model 1871 and Model 1901 pistols with the same type action.