All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His Slave and Messenger.
Islam sanctions reading Ruqyah on children. It was authentically reported that the Prophet read protective Ruqyah on his grandsons Al-Hasan and Al-Husayn, saying: "Aʻoothu bikalimaat-illaah At-taammah min kulli Shaytaanin wa haammah wa min kulli ‘aynin laammah (I seek refuge in the perfect words of Allah from every devil and every vermin, and from every evil eye)." Then, he said: "Your father (Prophet Ibraaheem [Abraham]) used to seek protection with them (these words) for Ismaaʻeel (Ishmael) and Is-haaq (Isaac).” [Abu Daawood]
Moreover, it was narrated that Abu Sa‘eed Al-Khudri said, "The Messenger of Allah, used to seek protection from the jinn and people’s eyes (i.e., the evil eye) until the Muʻawwithatayn (Soorah Al-Falaq and Soorah An-Naas) were revealed. When they were revealed, he recited them for protection and stopped reciting anything else."
Although Islam sanctions reading Ruqyah on children, the Ruqyah should not be confined to a specific time since the Sharee‘ah does not specify a particular time for it.
The questioner referred to preventing young children from going out in the evening. This has been mentioned in an authentic Hadeeth. Jaabir reported that the Prophet said: "When night falls (or when it is evening), stop your children from going out, for the devils spread about at that time. But when an hour of the night has passed, release them ...."
The Hadeeth does not command Muslims to perform Ruqyah for the children in that specific time, so one should not confine the performance of the Ruqyah to that particular time.
Al-Haafith Ibn Hajar wrote, “It is feared for children at that time because the impurity that the devils seek is usually present with them (children) at that time, and the Thikr (remembrance of Allah) that protects them from their evils is usually absent on part of the children, and when the devils spread out, they hang on to whatever they can hang on to. Therefore, it is feared for children at that time. The reason why the devils spread out at that time is that they are more able to move about at night than during the day, because darkness is more conducive to devilish forces coming together than any other time.” [Fat-h Al-Baari]
Allah Knows best.