As Snailboat mentioned in her comment, gogen-allguide.com has a good explanation for said etymologies.
Although, I should caution against analyzing Chinese moraically; Chinese is a syllabic language.
While {} and {} are etymologically borrowings from Chinese, they predate any of the three major classifications (ie, ), and are thus functionally nativized as indigenous Japanese lexemes.
The understanding is, at the time, Japanese had used a nasalized variant of /u/ (ie, /ũ/), which for /ũma/ and /ũme/ surfaced as [mma] and [mme]. Over time, this nasalization was lost, and thus, only the vowel quality /u/ surfaced, resulting in our modern /uma/ and /ume/.
(By the by, this nasalized /ũ/ also accounts for Chinese coda /ŋ/ ("ng") resulting in /u/, such as:
> /*kjiaŋ/ → /ki.ya.ũ/ → /kyou/