Generally, you don't run the reins through anything because that does limit their flexibility to be wherever you might need them at any time....but... if one is truly worried about losing them altogether out of the carriage, one can buckle a simple spur strap around the rein rail on the dashboard to create a big loop, and run one rein through the loop. That acts as a prevention to loss - as long as the driving reins are buckled, and the loop can travel the entire length of the rein rail to facilitate the use of the rein. But it does limit the free range of motion for the rein somewhat.
Some people will attach a "trailing rein" to the buckled ends of the driving reins - rather like a thin long leather lead rope attached - and that rein will remain in the carriage, dangling down to the floor with the bight held firmly under the driver's foot, so that if the driving reins go overboard, the trailing rein will keep the driving reins from being lost completely.
Now, if you ever do lose your driving reins, you better hope your animal responds to the verbal "whoa" because there ain't a lot you can do to retrieve those reins short of leaping out of the vehicle, or hoping the reins will run under a wheel to bring the animal up short (nasty on the mouth, tho) so you can leap out and grab them. If you have a groom, you can always put them down to run up to the horse (or pair) to grab the trailing reins.