Is there a rule for the movement of the "r" to the end of the word? Or is it moreso that there was some kind of intrusive "e" that separated the "-gr-" to form "-ger"?
The development is the result of:
- syncope of *-gros to *-grs (with syllabic r, sometimes transcribed as *r̥ following a non-IPA convention). In Latin, the Proto-Italic sequences *-ros, *-ris often show syncope of the vowel after *r, with the r becoming syllabic when preceded by a consonant (as in caper) but not when preceded by a vowel (as in vir, vesper).
loss of word-final s after r (most likely by assimilation *rs > *rz > *rr, followed by simplification of word-final *rr to /r/; for comparison, word-medial *rs became /rr/, as in terra)
vowel epenthesis that ultimately turned *r̥ into /er/.
I'm not sure about the ordering of the two last bullets.
See the following posts on Latin SE:
- Why do some 2nd decl. "-er" adjectives and nouns drop the "e" in the stem?
- Does /l̥/ in reconstructed Latin represent a voiceless (alveolar) lateral approximate or something else?
- Comparative Grammar of Greek and Latin, 1933, Carl Buck, §109, pages 99-100
Most sound changes don't have specific names.
A relevant point of comparison here is puer < *ph₂weros. This suggests that the masculine nominative singular thematic ending -os (as well as the vocative -e) was lost first, suggesting an intermediate form *agr which violates Latin phonotactics and so required the insertion of a vowel to make it legal. In most such instances Latin has i or u for epenthetic vowels, it seems likely that the choice of e here was influenced by some combination of analogy to puer (and similar words), the fact that an e already appears in the usual vocative ending, or some sort of vowel-colouring from the r (-ir and -ur being pretty unusual in Latin).