In 1997, a variant marketed as the Move Custom was introduced with larger headlamps and revised front styling, and was available in subsequent generations.

The Move was also offered with the turbocharged 659 cc JB-JL all-aluminium four-cylinder engine producing 64 PS (47 kW; 63 hp) and using the internal designation L602; this version was only available with front-wheel drive.

The second-generation Move was introduced in October 1998 and was also marketed in Malaysia with slight modifications as the Perodua Kenari.

This generation features a lightweight high-rigidity body structure with a reinforced underbody, improved suspension system, and a power mode-switching steering switch that allows drivers to change the control of the engine and CVT at the touch of a button.

In China, the FAW Tianjin joint venture between First Automotive Works and Toyota produces a petrol and electrical version of the Move.

Due to poor sales, Daihatsu did not export the third or fourth-generation Moves to Europe after the brand's departure from the continent, making it strictly a Japanese domestic market model.