If your trains hold value to you, be careful even with "professional" movers. Thefts happen, and movers don't necessarily care about damage when they're trying to fill cubic capacity on a truck.
I've moved cross-country twice and avoided the cost and questionable care of "professional" movers for most things. Trains and valuables are carefully packed, the boxes made trackable but not obvious as to their contents, and are among the first things loaded -- which puts them at the front of the cargo area and impossible to get to without taking everything else out.
If your stuff has to park for a while, professional moving companies will unload your stuff from the truck/trailer and pack it in a warehouse. Which leaves more opportunities for damage or loss.
On my current move, everything we didn't take with us and store ourselves is in moving pods -- the containers you load (or have loaded) and then lock with your own lock, at which point they're either transported to your new location or put wholly, still locked with your own lock, into climate-controlled storage until you call for delivery. My trains are packed safely and buried up in the front of one of those pods, safe from damage and thieves. I wouldn't have that kind of security with a "professional" moving company.
Moving companies are insured, which sounds great, but all it means is that when they lose or damage stuff, the insurance hands you money up to their limitation of liability. That's often insufficient, but you may not find out until it's too late.
If you can't keep valuables with you and may have to store them, I'd highly recommend at least packing them yourself and then loading/storing them with a good container pod-style provider. Much like with the big freight shipping containers, you retain more control over loading and access, and they can be stored safely as self-contained units.
