the comparison of sh and ch

[Jorge_Stolfi]

with regard to (I call it) "mutations" in general. It is noticeable that the ‘basic words’ if we assume that the letters are prefixes (or whatever without belonging to the base word) are subject to a vowel swap from "o" to "a" or vice versa. Not all words, but too many to ignore this phenomenon.
Are there any theories or topics on this? Is there a language or dialect from the Indo-European branch that has such a vowel exchange?

For Indo-European languages, I would have to check how common that is.

Offhand I can think of the Germanic umlaut changes when making a plural:

  das Buch -> die Bücher 

Also of stress and vowel quality changes when inflecting verbs in Romance languages like Italian "cantare" = "to sing":

  cAnto  "I sing"
  cantÒ  "he/she sang"
  cAnta  "he/she sings", or imperative "sing!"
  cAnti  "you sing", or subjunctive "that you sing"
  cAnte  subjunctive "that he/she sings"
  cantÀ  Rome dialect "to sing"
 
And then there is the verb "contare" = "to count", with "cOnto", "contÒ", etc. as above.  And also "cInto", "cInta", "cInte", "cInti", past participles of "cingere" - "to wear [a belt or diadem]"; and "cEnto" for "hundred", ...

And in English there is "cat", "kit", "cot", "cut", or "bat", "bet", "beet", "bit", "boot", "but", ...

Those English examples are just coincidences, but in other language families vowel change may be a fundamental feature.  In Semitic (Afroasiatic) languages like Arabic, Hebrew, and Ge'ez, a lexical root is usually three consonants, and it is inflected to form related nouns, verbs, adjectives etc by inserting vowels between those consonants, or adding short prefixes and suffixes:

From the root k*t*b = "things to do with writing":

  Book            kitāb    كِتَاب
  He wrote        kataba   كَتَبَ
  Writer          kātib    كَاتِب
  He corresponded kātaba   كَاتَبَ
  Writing         kitāba   كِتَابَة
  Write!          uktub    اُكْتُبْ
  I write         aktubu   أَكْتُبُ
  Office/Desk     maktab   مَكْتَب
  Library         maktaba  مَكْتَبَة
  ...
 
Turkish (and Hungarian too, IIUC) has a feature called "vowel harmony".  Simplifying a big lot, the vowels are divided in two classes "front" and "back"; and all vowels of a word, including all its affixes, must be of the same class.  Thus most affixes have two forms, one for each class.  For example, plural is made with a suffix that is either "-ler" or "-lar": "tepe" = "hill", "tepeler" = "hills", "dağ" = mountain, "dağlar" = mountains. In modern Turkish spelling (with Latin letters, created ~100 years ago) the suffixes are written attached to the word.  In an invented script they could be written as separate words; but they still would have to follow vowel harmony, and thus there would be at least two forms of each, differing on the vowels.

And then (ahem!) there are the monosyllabic languages of East Asia...

All the best, --jorge