When you say the driveway side is basically stone & dirt, I assume the drive itself is stone, either as stone pavers or loose rock. This probably allows enough water to penetrate the area and heave the channel system. To keep the existing channel in place you'd have to pour a deeper layer of concrete on the bottom & anchor the channel down, or pour a minimum of a 2' concrete apron on the drive side of the channel grate.
The stone/dirt alters my initial thoughts, so I'd instead consider using a 4" perforated S&D pipe, socked with a loose-weave mat, like a weed barrier, laid in the bottom of the trench, then surrounded in pea gravel. Cap one end unless it drains to both sides.
If you think much dirt & silt will be washing in with water, scratch the sock & just use the pea gravel; the sock would silt in, while the gravel will allow it to pass on. Be sure to use a pea gravel or similar; basically rounded rocks with no flats, and not so small they would easily lodge in the pre-fab holes in the S&D.
If you channel the water beyond the sides of the drive, switch to solid S&D at the sides for the remainder of the discharge line.
That's my thoughts on this one, David; hope it gives you some inspiration.