If someone is suffering the Grappled condition and is then moved by another means what happens to the person providing the Grappled effect?

In a situation the fighter had grappled with the scary bird monster, another character who was "behind" the fighter casts Lightning Lure. The creature failed its Strength save and so was pulled towards the caster. We allowed that the fighter would pivot to allow the Lightning Lure to pull the creature.

Was this a fair reading of the rules? As Lightning Lure only has a range of 15' it seemed that (if using squares) that it would go from 
Monster|Fighter|space|space|Caster 
to
Fighter|Monster|space|Caster

I don't think there are additional considerations for Thorn Whip's longer range, but answers could include any such details.