They can carry far more cargo than any other form of transportation and need essentially nothing in terms of infrastructure between the launch and the landing.
Payloads were close (for example, 100 people and their cargo) in the 1930s.
Proto-type airships with 60-75 ton payloads are under serious investigation for military and commercial use today, and the technology scales well with 200 and 500 ton payload versions on the drawing boards.
The biggest load by a single helicopter at once is 105 tons. But, with several, very large helicopters going a fairly short distance, you could probably carry a bigger payload, although I don't know that anyone has ever done that with quite such a heavy load.
625 tons is no big deal for a cargo ship or barge, and also long as the origination point and destination both have connections to a port or even a navigable river, you are set.
Some Assembly Required
Just because it weighs 134-625 tons full assembled, doesn't mean that you can't take it apart into manageable sized pieces and put it back together at its destination. A semi-truck load or C-5 Galaxy transport can probably manage 70 ton loads, which could be as little as two or three loads for the Amtrak and perhaps 10 loads for your Big Boy.
Whole castles and towers have been shipped across the Atlantic in this manner.
Build A Temporary Rail Line To Connect The Lines
Build a one time use rail line to connect the points and then tear it out.
Adapt It For Road Use
Build an adapter (or an oversized flatbed truck) that allows the train wheels to be used on a road temporarily that can be removed once the destination is reached. The adapter doesn't have to be fast, fuel efficient when used, or durable and can be built so that it only operates at optimal temperatures and/or weather conditions.
This (together with barges) is basically how large blocks of stones were put in place for the pyramids.