Most designated marksman rifles are based on modified designs of an assault rifle or currently issued by a nation's military, or on a battle rifle that was formerly issued. The ammunitions used are often of the same caliber as that of the machine guns within the same combat unit, typically a fully powered cartridge such as the 7.62 mm NATO. For example, battle rifles such as the M14, FN FAL, AR-10 and H&K G3 were largely replaced during the 1980s and 1990s by modern assault rifles firing the 5.56 mm NATO intermediate cartridge for standard riflemen, but many were accurized and retained as DMRs. Conversely, some nations have also built rifles that were designed for the designated marksman from the ground up. Examples include the Soviet SVD and Chinese QBU-88.
All designated marksman rifles in use today are semi-automatic, some with select fire. Sniper rifles are generally bolt action rifles, but can also be semi-automatic.
I could conclude that DMRs are essentially the middle-ground between assault rifles and sniper rifles. They are capable of covering long ranges but not sniper levels of range. Unlike sniper rifles, the semi-auto mode of DMRs result in higher fire rate, and they used to come in higher capacity magazines.
You are absolutely correct. The H&K G28 is the military variant of MR308 semi-automatic competition rifle
