Mortarless bridges are more challenging to build than standard, mortared bridges.
The Advantages of Mortared Masonry
Mortared masonry is more forgiving then mortarless stonework — the mortar fills in the gaps between stones, forming custom-shaped stones that perfectly fit into the joints. This makes the mortared masonry very strong. Well-built mortared masonry also effectively seals out water, which means that stones will be less likely to be affected by freeze-thaw cycles.
All that said, however, nothing can compare to a well-built, mortarless stone arch bridge when it comes to beauty and cost. And, with proper caution, mortarless masonry can be built to hold together and defy frost action.
The Key to Building with Mortarless Masonry
The main thing to determine when building a mortarless masonry bridge is in what direction the stones will try to move. If the stones have an inclination to slide in a certain direction, frost action will tend to start them moving in that direction.
Consider, then, the case of a stone in the spandrel walls of the bridge, that is placed on a wedge-shaped stone below and therefore leans slightly out of the bridge. Over time, the stone will slide out, and fall out of the structure.
Given the same scenario, but this time with the stone leaning into the fill, the result will be stable. Why? Because the fill is pushing out, but the stone is pushing in, and an equilibrium is established.
Therefore, dry-laid spandrel walls should ideally be gently leaning into the fill. Dry-stacked walls (no fill) should typically consist of two semi-independent walls leaning into each other; the two leans cancel each other out, resulting in a stable wall. For more information, please see our article on Mortarless (Drystack) Retaining Walls.

Dry-laid arches are not overly vulnerable from a frost standpoint. The whole arch is pushing all of the individual arch stones together and frost action will have little effect on it. Foundations should likely be built as a dry-stacked wall; however, as the structure is underground, the dirt all around the stones will tend to hold them in place regardless. Also, the weight on the foundation tends to increase the friction between the stones, making the foundation stones rather disinclined to move.

The hardest stones to keep in place are the top capping stones. Not only do they have little weight on them, but they are prone to impacts from bridge use. Making these stones large helps. Building up the fill to just reach and maybe even cover the tops of these stones will also help hold them in place. Dirt fills can have a surprisingly stabilizing effect on the stonework if done correctly.
If the dry-laid masonry is made of large stones, it is difficult for frost to move the stones around. Huge, half-ton blocks of stones are not likely to slide around in a hurry. Also, as frost’s power is related to water in the masonry joints, it follows that, if the structure is well drained, it will hold together better. Of course, as a stone arch bridge is usually on water somewhere, there is probably going to be some moisture in the structure.
One important thing to remember about dry-laid masonry is to use chips and shims of stones in the joints to maximize friction between the stones, and to keep them from rocking and moving. These small fragments of stone play a key role in holding the structure together. And, of course, follow the standard masonry basics.