"to fall behind" is the process of becoming further away from something or someone in front of you
After you have "fallen behind", you may then "lag behind" them or it. "Lag behind" refers to continuing in that position.
"I was running the race neck and neck with my friend for 5 miles. During the next mile, I fell behind. I then lagged behind my friend for 3 miles. I then caught up with him."