All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
We are not aware of any religious evidence which proves the permissibility of moving (one’s head forward and backward) when reciting the Quran. Rather, among the desirable etiquettes when reciting the Quran is to be tranquil and serene. However, the movement which you mentioned in the question contradicts this and this is a kind of imitation of the Jews. Indeed, Shaykh Bakr Abu Zayd stated in his book ‘Innovations of recitation’ that the scholars of Andalusia strongly objected swaying, rocking and moving when reciting the Quran, and stated that it is a Jewish innovation which infiltrated to some Oriental reciters of the Quran. There is no authentic narration that the righteous predecessors of this nation did so. Therefore, one should avoid doing so, but if this happened unintentionally, then one is not sinful.
Finally, it should be noted that the people who do Da’wah should advise each other, because advising each other is one of the requirements of the Islamic brotherhood, but this should be with wisdom and good admonishing.
Allaah Knows best.