1) Alter 2) Bemire 3) Change 4) Colly 5) Cover with smut 6) Dirty 7) Grime 8) Make soiled, filthy, or dirty 9) Modify 10) Smudge heavily 11) Soil 12) Soil with oil 13) To make grimy 14) To make something dirty 15) To soil Found on https://www.crosswordclues.com/clue/begrime
Be·grime' transitive verb [ imperfect & past participle Begrimed ; present participle & verbal noun Begriming .] To soil with grime or dirt deeply impressed or rubbed in. « Books falling to pieces and begrimed with dust. ... Found on http://www.encyclo.co.uk/webster/B/32