Game flicking chequers knocking opponents off
I just googled &quot;flicking Draughts game&quot; and the first result was a wikipedia page on a game called Chapayev
Details from the article include:-

The game requires a checkerboard and checkers, eight pieces of each colour.

Which matches your question.

During the first round, white pieces are placed on the first row, and
black on the last.

Which fit requirement of &quot;checkers on opposite sides of the board&quot;

A player takes their turn by flicking one of his pieces with the index
finger to shoot it at one or more of the opponent's pieces.

Which again matches your description

The game continues until only one colour remains on the board, winning the round.

Which matched the victory condition you describe.