Hold a hairdryer set on medium heat several inches above the vinyl tiles that have shifted. Apply heat with the hairdryer by moving it slowly back and forth over the tiles for several seconds.
Lift up gently on one of the tiles that has shifted at the corner. Apply more heat with the hairdryer if you meet resistance. Pull up any other tiles that have shifted in the same manner.
Scrape the old adhesive off the subfloor with a paint scraper. Wipe the paint scraper with a paper towel after each pass and continue scraping the subfloor until you have removed the old adhesive. Gently scrape the old adhesive off the back of all shifted tiles in the same manner as on the subfloor.
Wipe the subfloor and the back of each shifted tile with a damp rag. Let the subfloor and tiles dry for 15 minutes. Brush mastic on the subfloor with a paintbrush then replace the tiles flush against each other.
Use a rolling pin to apply pressure to the tiles in a back and forth motion. Cover the tiles with wax paper and set a heavy object on top of them for 24 hours.