Einsatzgruppen
The Einsatzgruppen did these jobs in many ways.  Sometimes, they would make lists of people who they thought were enemies, and would then murder them.  Other times, they would perform massacres of thousands of people over a few days.  For example, at Babi Yar, one Einsatzgruppen battalion killed 33,771 Jews in two days; in the Rumbula massacre, another killed about 25,000 people in two days.