its not an item so no effect 

the bag says that it only has an adverse effect on being inside an item that is a dimensional space

Quote from DragonRiderHN >>

What happens if the rope is moved? Can it be moved? What if the lower end of the rope is destroyed (i.e fireball) or cut away? Does the space fall until the rope is once again touching the ground?

Quote from Krodothlhren >>

My Question is how does this spell work when cast while on a moving object like a ship or a wagon?

Quote from DavrosMandrake >>

Been through the thread looking for guidance and on the subject of Bags of Holding I think the question should be:

"What happens if you Open a Bag of Holding inside the dimensional space?"

However, that is not why I am here, instead its to see what is thought on the subject raised by one of my players "If I were to cast rope trick in the back of a moving cart what would happen? Would it be anchored to the space in the cart or would it remain stationary relative to the ground. If relative to the ground, would it being cast in the enclosed cabin of a wagon make any difference."

The same would apply to casting the spell on the deck of a ship or in one of the cabins, or even in the compartment of a lightning rail.

 
As a DM, not a rules lawyer (so no references, just house ruling and fun with friends) :

I would personally allow the extradimensional space to move with the transport as long as:
One end of the rope is tethered via the Spell at one end(inside the Rope trick, the rope's end sits in the middle of the entrance) and the other would have to be tethered in the casting plane (to the cart, ship, lightning rail car).
IF THE ROPE WAS UNTETHERED from the CASTING PLANE - The movement the having no momentum would stop being transported (With the transport moving away with no damage, even through walls and creatures may get in the way - it's not in our space), with no one aware within the "Space" unless looking through the entrance.
IF THE ROPE WAS UNTETHERED from the extradimensional space(Rope put in a backpack), dispelled by caster, Time expires - The contents of the extradimensional space is dumped back into the Casting plane, where the entrance resides.
Known problems: 
It makes the rope trick a 'balloon on a string' with a party inside: so I'd say it cannot move unless it is tethered to a structural part of the transport (towed/secured). Otherwise your ruling a free 1 hour immovable rod with the breaking strain of the rope. 

My question is this:

Could you put the party members in the interdimensional space, and have a party member with Boots of Speed, grab the end of the rope and Take off running for the duration of the Boots of speed, would that Interdimensional space move with the person running with the end of the Rope. 

It doesn't say anything about the Space being able to move if the Rope end is moved?  I of course Say YES...