Asalam o Alikum,
Dear Sir/Ms.
1. I would like to know about the status of the children in case of Misyar. Who is going to take the responisibilities of the children (Nan-Nafqa).
2. Does the children have any share in the propoerity of the parents.

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.

The Misyaar marriage that fulfils the conditions and pillars of a correct marriage is a valid marriage and so it leads to the known effects of a normal marriage, except that the wife loses some of her rights.

If there are any children from such marriage, they are the children of the husband and he is obliged to spend on them and accommodate them, and they inherit from him if he dies before them and he inherits from them if they die before him.

For more benefit, please refer to Fataawa 82104 and 85593. 

Allaah Knows best.