"More and more" typically has a connotation of "increasingly" or "additional". "Further" is slightly different, but very similar: "past or beyond the current point". Additionally, "further" is more likely to refer to something that cannot be "chopped up" into individual parts (e.g., "further analysis", "further consideration", "further thought"), whereas "more and more" may refer to sets of discrete objects (e.g., "more and more gumballs", "more and more thoughts").
Due to the way it sounds (and word usage), I would avoid saying something like "plagiarism more and more becomes more common" and stick to something like "plagiarism further becomes more common". However, I think this wording is still awkward. I don't think the use of two comparatives is needed. Instead, one might say "plagiarism becomes more and more common" or "plagiarism becomes increasingly common" or even "plagiarism continues to become increasingly common".
An instance where "further" belongs might be: "plagiarism is most severe when students have internet access, further complicating the modernization of schools".