Move auto to a dolly to take to shop?

Using an automotive dolly can be risky, If the vehicle is front wheel drive then dolly the front, if it is rear wheel drive then dolly the rear. If it is 4 wheel drive and late model with no neutral shift position dolly the front and remove the rear drive shaft at the rear axle. If the vehicle is older 4 wheel drive with neutral selection for the transfer case then simply shift to neutral and dolly the front axle. Most transmissions whither automatic or manual will be damaged if the engine is not running while the vehicle is in motion.
the engines crankshaft or the transmissions input shaft must turn to lubricate the transmission unless the transmission is automatic and has a rear pump that can lubricate the transmission if the transmissions output shaft is turned in normal rotation for forward direction of travel. This is the only way to to assure there is no damage to the drive train. Be sure to follow all recommended tie down procedures. remember the dolly rotates under the vehicle so one or the other of the wheels must be able to move forward and back to allow for corners. Try to always choose a path that allows forward direction unless you are an expert driver with experience and able to back a farm wagon type axle trailer (like covered wagon) going backward will be a challenge even for someone experienced pulling a bumper pull trailer.
To get the vehicle onto a dolly have some friends help push or use a cable puller to move the vehicle onto the dolly.