Most fish move as swimmers in bags. A bristlenose pleco moves as a hermit in furniture. He spends his days sealed onto one piece of driftwood or inside one cave, emerges at night to graze, and measures home by that hiding place. Plan the move around the wood and the pleco largely takes care of himself; plan it around netting the fish and you will spend a stressful morning trying to detach an armored, suction-cupped animal that has no intention of cooperating.
Move the hiding place, not just the fish
If his driftwood or cave fits in a sturdy lidded tub, lift it gently into tank water in that tub and check whether he is riding on or in it; very often he is, and the move becomes a matter of carrying the tub. If he is elsewhere in the tank, guide him with a container rather than a net, since his pectoral spines tangle in mesh. Never pack him in a plastic bag: the spines and the bristles on a male's head puncture bags, and a slow leak discovered at the destination is a crisis. A hard tub with a secure lid is the only sensible vehicle.
Prepare his corner of the new tank first
Set the destination tank up before he travels, with his cave or wood positioned in a shaded corner away from the brightest light, conditioned water at a steady temperature in his broad comfort range, and his old filter media moved wet so the water is safe. Fast him for a day before the move; plecos produce more waste than their size suggests, and a clean tub matters over a long drive. Keep the tub dark and in the car cabin, and resist opening it en route to check on him. Darkness is his comfort, and every peek is a disturbance he did not need.
Let him emerge in his own time
At the new home, acclimate the tub water gradually to the tank, then place the wood and let him stay in or on it as it goes in, rather than shaking him loose. Lights off for the rest of the day, and after dark, sink an algae wafer near his cave. Then be patient in a way other fish do not ask of you: a bristlenose may not be seen for days after a move and still be feeding at night. Judge him by indirect evidence, wafer gone by morning, droppings appearing, wood rasped clean. Only worry if over the following couple of weeks he looks visibly thinner with a sunken belly, which suggests he is not eating at all; check the water readings and his access to food against faster tankmates, and seek an aquatic veterinarian's advice if the weight loss continues.
A KinsLe Family Sharing note on his profile, recording where his cave sits and the plain fact that not seeing him is normal, keeps every helper in the new house from helpfully pulling his home apart to look for him during the first unsettled week.