More precisely: a matter-dominated or radiation-dominated universe will move further away from flatness as it expands (unless it happens to have exactly the critical density and be perfectly flat).
Our universe is currently dark energy dominated, so it is (very slowly) moving closer to flatness as it expands (assuming it isn't already perfectly flat).
According to inflation models, our universe was also (effectively) dark energy dominated (the inflaton field acts like dark energy) during inflation, which (rapidly) drove the universe very, very close to flatness, close enough that the period of 10 billion years or so after inflation ended when the universe was radiation and then matter dominated did not move it far enough away from flatness for us to detect the difference.