On'yomi and Chinese: how sounds correlate
In almost* any discussion of kanji usage in Japanese, do not use the Mandarin pronunciations as any kind of guide to the Japanese pronunciations.
(* The exception is any discussion of recent borrowings from Mandarin, like [你好]{ニーハオ}.)
Much as Japanese has changed a lot in the last 1500+ years, so too has Chinese changed. When exploring Japanese on'yomi pronunciations of kanji, you'll want to look at the Middle Chinese readings, not the modern Mandarin, and you'll need to be aware of the Old Japanese pronunciations as well.
Here's an exploration of the five kanji you listed. The Middle Chinese are the Zhengzhang reconstructions, as shown in the English Wikipedia entries for each kanji (as linked).
Kanji |
Middle Chinese |
Old Japanese |
Middle Japanese |
Modern Japanese |
波 |
/puɑ/ |
pa |
fa |
ha |
比 |
/piɪX/ |
pi |
fi |
hi |
不 |
/pɨu/, /pɨuX/, /pɨut̚/ |
pu |
fu |
fu, bu* |
部 |
/buoX/, /bəuX/ |
bu, pu* |
bu, fu* |
bu, fu* |
保 |
/pɑuX/ |
po |
fo |
ho |
The items marked with * show readings classed as kan'yōon, which may be the result of influence from different regional or dialectal variations within Chinese.
The initial consonants in Middle Chinese and Old Japanese are a match. The vowels are mostly a match, provided that we make certain allowances for diphthongs (two-vowel sounds) in Chinese flattening into monophthongs (one-vowel sounds) in Old Japanese.
Sound shifts from Old Japanese to Modern Japanese
Historically, all of the はひふへほ kana in modern Japanese were pronounced more like //pa//, //pi//, //pu//, //pe//, //po// in Old Japanese. See this post for a fuller explanation of the details.
Your question
Now that we have the background in place, we can look at your question more fully.
Are there evolutions or relations of why and how the Chinese sounds (Mandarin or old Chinese) B sounds evolved to the Japanese h or f sounds for these Hiragana?
- The modern Mandarin intial //b-// was formerly //p-// for four of these characters.
- The modern Japanese "h or f sounds" were formerly also //p-//.
The one outlier is 部, which is reconstructed with initial //b-// in Middle Chinese. I note that modern Min Nan has initial //p-//, and subjectively, I've noticed that the on'yomi for many characters seem to correlate most closely with this variety of Chinese. I suspect that proto-Min Nan may likewise be the source of the unvoiced //pu// reading in Old Japanese.
Addendum
Above, I was looking at on'yomi and I didn't address 部 used as the source for the kana へ.
Chinese of any dialect or historical stage is irrelevant in this case. へ is not a Chinese-derived on'yomi for 部, and is instead kun'yomi, from the native Old Japanese word that happened to have roughly the same meaning as 部. This is why you won't find any clear correlation between any Chinese pronunciation for 部 and the Japanese kana へ, even though the glyphs (character shapes) are historically related.
Please comment if the above does not address your question and I can edit to update.