What is the ruling on bringing children to mosques in Western countries to raise them to participate in rituals, despite the disturbance this causes to others, and is the hadith "Keep your children away from your mosques" authentic?

The aforementioned Hadith (definition) prohibiting children from entering mosques is weak. Other Ahadith indicate the permissibility of their entry, such as the Prophet, peace and blessings be upon him, lightening his prayer upon hearing a child cry, and carrying Umamah bint Zaynab while he was praying. There is no doubt that bringing children to mosques has great educational benefits.

If the children are discerning (mumayyiz), they should be taught the etiquette of the mosque and enjoined to adhere to it. If they do not comply, their guardians should be informed to discipline them. As for children who are not yet discerning, their presence is allowed if they can stay with their guardians without causing disturbance or defiling the mosque. If this is not possible, they should be prevented from entering the mosque during prayer times and sermons.