No, you cannot directly swap the transmission from a 2001 Dodge Durango 4.7L to a 2003 Dodge Durango 5.9L without significant modifications, and it's generally not recommended. While both might use automatic transmissions, the 5.9L engine produces considerably more torque. The 4.7L transmission is not designed to handle that increased stress and will likely fail prematurely. Even if you could physically bolt it up, the torque converter and internal components wouldn't be suitable for the 5.9L's power output.
You'd need a transmission specifically designed for the 5.9L engine. Using the wrong transmission could lead to catastrophic failure, potentially damaging the engine and other drivetrain components.