How to Run Wires in Existing Construction past horizontal stud/fire-block

Well,  I have a few questions??
1) Will you be using the existing wire still or will that be replaced once the new wire is installed?
2) Is the thermostat on the 1st or 2nd floor?
3) You can try to run a home-run through your attic if you are attempting to access walls on the 2nd floor. You wont be able to access 1st fl walls from the 2nd floor no matter how much you try without causing damage. a GOOD RULE of thumb is: fishing walls on the 2nd floor is usually easier from the attic. Fishing walls on the 1st floor is typically easier from the basement.
4) The simplest solution would be to use the existing wire to pull through the new wire(s) right from the back of the "Existing" thermostat and pull DOWN towards the basement..... Or vise versa.... Pull up from the basement.....

I threw a few Ideas out there... I hope you can use one. Let me know if there are any other questions you may have.....

For what its worth:
Typically your standard thermostat wire is insulated and thicker (Diameter wise*) than your typical ethernet, CAT5, etc...Its all 8-condutor but its alot thinner than the thermostat wire...

To answer The Warlock's questions above...
1.  Yes, I will be using the existing wire.  The Cat5 I need to run is additional to the normal thermostat wires.
2. The thermostat is on the 1st floor.  
3. Actually, my home is 1.5 stories.  The location of the thermostat is in the part of the house that is just a single story, so I could access it from either the basement or the attic.  I could go up through the attic, but then I would need to run down through another wall somewhere to get back to the basement where my hub is.  I was trying to avoid having to do that, since it could lead to further complications.  But right now it's either that or busting a hole in the wall - unless I get some other suggestions here.
4. That's what I did when I replaced the 4 conductor with the 6 - I used the 4 to pull the 6.  But in this case, the 6 needs to stay there and I need to add the cat5.

He asked for, "easiest/best/cheapest way to run these wires".

IMHO, cut a hole in the wall, pull the wire out, round out the hole to make it larger, reach in grab the wire and push it through the hole - is pretty easy and cheap.  Yeah, a little labor repairing the wall.  But meets the criteria.

I really think that calling the patch job 'a little labor' in glossing over just how much extra work this creates to drop a couple wires down the wall.  Probably why the poster asked for other suggestions.

While patching large holes in drywall is not 'hard' it is not 'easy' to do well - meaning done so it does not show.  Unless the tools and materials are handy it is not necessarily cheap either.

Here's a link for the drill bits - $25-30 range,  a local store would likely be a bit more.  Well worth the cost for the mess it saves.

http://www.lashen.com/vendors/BES/Fish_Bits.asp