Without more context (the nature of XYZ, what 'promotion' means in this sentence, whether the research is capable of 'promoting' XYZ or not), I would understand the difference between your sentences to be:

I heard about your research promoting XYZ = your research promotes XYZ (you've done some research deliberately to promote XYZ)

I heard about your research on/into the promotion of XYZ = your research researches the promotion of XYZ (an academic study researching promotion techniques of XYZ?)

Both sentences are correct, grammatically, but as I said in #1 (confirmed by CM2DD in #2), they mean different things. (Although I'm not totally clear what it means to 'research the promotion of multicultural relationships'!)

I heard about your research promoting multicultural relationships = you have done some research in order to promote multicultural relationships / the purpose or goal of the research is to promote multicultural relationships

I heard about your research on/into promoting multicultural relationships* = you have done some research about the promotion of multicultural relationships. The purpose of the research is analysis, not promotion.

*I actually prefer research into the promotion of...here but that might be personal preference.