The Model 1888 was loaded with a 5 round "en bloc" clip and was chambered for an 8x57 round using a bullet with a diameter of .318 inches, called in the civilian world the 8x57I.* It was found that the rifling was subject to corrosion and erosion and the rifling was deepened even though the bore diameter (.311") was not changed. Later, the same case was enlarged at the neck to take a .323" bullet. The civilian name for the new round was 8x57IS. The military, in the perverse ways of the military, changed the name of the cartridge from 8mm to 7.9mm, even though the bullet was actually larger!
In any case, the old rifles were converted simply by reaming the chamber, thus enlarging the chamber neck to prevent high pressures, and altering the magazine to use the new ["stripper"] clip, so that standard 7.9 ammunition could be used. Later, many of those rifles were given to Germany's ally, Turkey, and some remained in service for many years in that country.
*The I stands for Infanterie (pronounced Infan ter' ee); but at the time, there was no "J" in German. In other words the German alphabet had only one letter between "H" and "K", though it could be pronounced either as an English "I" or as a "Y". Many non-German companies adopted the "J" instead of the correct "I" for the ammunition, and that has stuck, so the cartridges are also called the 8x57J and 8x57JS.
Jim