When I shift my SRAM Eagle AXS Transmission repeatedly, the rear derailleur seems to take time to catch up and shift. Why is that?

With Eagle Transmission, you can shift under full load. Cassette Mapping technology controls shift speed, so that the chain always aligns with the rigorously designed shift ramps on the cassette to ensure both up and downshifts happen at precisely the right time for the smoothest shifting. Aligning the chain with these shift ramps on a T-Type cassette ensures that the chain remains firmly engaged with the X-SYNC tooth profile for smooth unbroken power transfer while shifting. Don’t wait for the shift, just keep pedaling under power!