Periodicity of algebraic $K$-theory in high enough degrees with finite coefficients

Question

Given this it seems that higher algebraic $K$-theory and the etale one coincide in high enough degrees. The etale $K$-theory with finite coefficients is also Bott inverted $K$-theory, so it should be periodic with even period. Given these, it implies that mod $l$ algebraic $K$-theory should be periodic (they repeat with some even period) in high enough degrees. Is this observation correct, if so are there any interesting examples that these have been calculated and it is possible see this periodicity? (I am mostly interested in smooth varieties).