Moving a 5,000 vinyl LP collection safely

I recently moved from the lower Hudson Valley to Austin, TX, with approximately the same number of LPs. I packed them myself in double walled boxes that I got in bulk from one of the specialty suppliers, Bags Unlimited. I numbered each box and had a word document (i'm not great with spreadsheets) inventorying them, some specifically by title, others by catalog (where I owned a lot of a particular catalog). The movers taped the boxes, and hauled them from a third floor to the truck.
This was in the dead of winter, so heat wasn't the issue, but water slop was. The records were taken to storage in Conn. and in mid-April, hauled to Texas. I don't think 80 degree F temperatures are a big deal.
 Since commercial movers are what they are, no matter whether "white glove" or not, it may be worth talking to your insurance broker for some additional insurance during the period the records are in the mover's hands. Not that this is a complete solution, but it may give some additional comfort. 
My records arrived unscathed. 
Hudson Valley is beautiful in autumn and spring. Sounds like you are going north of Bear Mountain, which is approximately 50 miles north of the city. We were near Piermont. 
Good luck, time consuming, tiring but you'll get it done. 

Now that that's solved, what town are you moving to?

I'm moving cross country with about the same amount of LP's. 
I have heard two schools of thought on packing them in boxes.
Some guys say to pack them LP's standing up and other say to pack them flat.
When I worked at a record store many years ago, the records would always ship flat.
Any thoughts on this?

Also consider that the records would have to reach the 80 degree mark or whatever temp you are worried about.  Just because the air temp in the transport vehicle reaches X the heat has to penetrate the cardboard which is double walled with an air-space and then warm the records which are further insulated within their cardboard jackets.  The mass of vinyl would have to cook a LONG time before even the outer ones gained enough heat to warp.  Worry about an accident instead as that is far more likely to damage your collection.  Now a move where it might take weeks to arrive is a whole 'mother kettle of fish.