A carrier a dragon cannot argue with
Bearded dragons are strong, clawed and curious, three reasons the carrier matters. A rigid pet carrier or tub with a locking lid, air holes and a lining of paper towel or shelf liner for grip contains the scratching that shreds soft carriers. Size it so the dragon can turn and flatten comfortably, with a low branch or rolled towel to brace against. Put the dragon in at the last moment, once the car is packed, and expect a minute of scratching at the sides before it accepts the arrangement and settles.
Temperature: shade beats heat
Owners worry about cold and cook their dragons with kindness. A healthy dragon tolerates a few hours at ordinary room temperature perfectly well; what it cannot escape is a carrier in direct sun, where temperatures climb past anything a basking spot should reach and keep climbing. Shade the carrier, run the car's ventilation as you would for yourself, and in winter warm the car first and add a wrapped heat pack outside one end on long runs. A dragon gaping and flattened like a pancake is too hot; one that is dark and still is usually just asleep or annoyed.
Stops, snacks and water on long drives
On drives long enough to need stops, the dragon's needs are simple. Offer water by dripping it on the snout at a stop, since many dragons will not drink from a travelling bowl but will lick droplets. A few greens can be offered on a long day, though many dragons refuse food in transit and lose nothing by it. Keep stops short and the carrier shaded, and never leave the dragon in a parked car in warm weather while you eat: the timescale for a dangerous car interior is minutes, not the length of a meal.
Out of the car at rest stops
If the dragon comes out at a rest stop, it wears a harness or stays in arms over grass, well away from the car park's traffic and dogs. Dragons startle at shadows from above, birds real or imagined, and a startled dragon is astonishingly fast across open ground. Most trips, the kinder choice is no outing at all: the carrier stays shut, the dragon sleeps, and the adventure happens at the destination. Children should hear the rule before the stop, not during the scramble.
Settling at the far end
Arrive and set up before you socialise the dragon. Get its enclosure, or the travel setup, holding the right gradient, basking spot at the level it knows from home, water offered, and let it warm through and survey the new room from its basking place. A dragon that basks, then eats that evening or the next morning, has travelled well. If illness or injury interrupts a trip far from home, the visiting vet's first questions are all history, diet, lighting and past problems, and a dragon travelling with its records in a KinsLe Pet Passport packet answers them from paper, not from a tired owner's memory.