The 2001-2003 Toyota RAV4s are notorious for transmission issues. In fact, the 2002 RAV4 is one of the worst models for problems with shifting, jerking transmission, and ECM failures.
These problems were so bad that Toyota was forced to extend its warranty from five years-60,000 miles to 10 years-150,000 miles, and coverage for the ECM and transmission. There were other reports of hundreds of cases of transmission failure. However, most of these can be traced to the ECM.
More recently, the 2019 Toyota RAV4 also suffered a similar problem. Many owners have complained of the car’s hesitation when they try to accelerate from a slow roll or lower gear. Resulting in lurches when they try to speed up. It’s no surprise, though, seeing as that model has seen seven recalls since it was released.
Just like the second-generation RAV4, the likely culprit is the ECM. Take the vehicle to a verified service center and have them reprogram or replace the unit.