I think it would be a mistake to let movers handle your trains. I know a number of people who have had valuables disappear during a move.
Thanks to the media, there are plenty of folks who think Lionel trains are valuable. They certainly aren't traceable.
My suggestion: pack your trains yourself, and seal the boxes. Do not mark the boxes with the contents, just a number.
If you are moving a short distance, move them yourself, you may need to briefly put them in a storage unit.
Bubble wrap is bad news. If left in contact with your trains for an extended period, it can leave marks in the finish of your trains.
When I pack trains, I put them in a food grade plastic bag. Don't seal the bag, you do not want to trap moisture. If I have a storage box, they then go in the box, if not, they get wrapped in newspaper.
I use generic boxes, lately it's been 800 count sports card boxes. If there is a lot of leftover space, I use some crumpled newspaper to fill it. I was planning to visit a Sports Card shop to pick out some smaller sizes in the next couple of days.