I have a doubt about Lailath al-Qadr. Will the Lailath al-Qadr be different at different places? For example if 27th night is here in India it will be morning in America. Angels will leave the mosque in the morning in India. At this time it will be night in America. Then will the Angels come down again to mark Lailath al-Qadr in America? Hope you will answer this question.

Praise be to Allah, the Lord of the World; and blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

As a rule, a Muslim has to believe in al-Ghaib without any form of questioning. Al-Ghaib literally means a thing not seen. Al-Ghaib according to Islam, comprises to believe in: Allah, Angels, Holy Books, Allah's Messenger, Day of Resurrection and al-Qadr as well as what Allah has informed about creation, and about Paradise and Hell.
As mentioned above, we are to believe in al-Ghaib since the human mind can not perceive a manner in which Angels descend in the night of al-Qadr or any other nights, or even the manner in which Allah, The Great and Almighty, descends to the heaven of this world at the last third each night. Allah alone knows that matter since our minds, in their human nature, can not comprehend it. And if we do not know or see something, this does not mean it does not exist.
It is our duty, as true Muslims, to say: we believe in what Allah has informed us of in His statements as well as in what His Messenger stated and implied in his sayings.
Allah Praises His servants in the Qur'an saying: "This is the Book (the Qur'ân), whereof there is no doubt, a guidance to those who are Al-Muttaqûn [the pious and righteous persons who fear Allâh much (abstain from all kinds of sins and evil deeds which He has forbidden) and love Allâh much (perform all kinds of good deeds which He has ordained)]. Who believe in the Ghaib …" [2:2-3].
Furthermore, it is true to say that the descent of Angels is continuous, i.e. their presence is as successive as the passage of time in each and every place.
Allah knows best.