Cendrine,
Thanks for reading! In terms of the Quran, it lacks the metrical quality of classical Arabic poetry, so its recitation differs greatly. But it still follows a certain poetic form called saj', in which the end of each line is similar in rhyme and rhythm.
Take, for example, the following final words from the first five lines of surat al-zalzala:
zilzalaha
athqalaha
ma laha
akhbaraha
awha laha
Not only do these terms share a common rhythm, they also contain multisyllabic rhymes, instead of the monosyllabic rhymes of classical Arabic poetry.
So, although the verses of the Quran don't conform to a meter, the conventions of rhythm and rhyme are preserved in the final word of each line.
The movement you refer to when reciting the Quran may therefore have more to do with the reciter's concentration, similar to how Jews can sometimes sway back and forth when praying.