Brother trany in vitz is K410. It's very sensitive to the oil being used in it. Generally mechanics don't have knowledge and they replace it with ATF which slowly kills the transmission. Toyota CVTF must be replaced every 40k Kms. Transmission once screwed for any reason can't be repaired and replacement is expensive. Before buying vitz do check the trans fluid it's running. And whether it was replaced at correct intervals. It's replacement procedure and level is very precise as well. Car has to be level within 1°. Trans oil temp must be between 36 to 45°C. Any sensible expert mechanic wud know all this.
Now the engine. Vitz has 1KR-FE 3 cylinder engine. If oil isn't changed in time or engine runs tap water in radiator rather than coolant, the damage is done. There wud be sludge that causes multiple problems. Mechanics generally remove the cat converter and drill tiny holes in throttle plate to adjust idle rpm. If these sort of alterations have been done... Stay away... It'll a continuous pain in the neck.
That's my limited knowledge and experience. Experts can further refine and explain.