Is it permissible for a girl to marry and move to live with her husband in another country, leaving her mother and brother (19 years old), with the mother's knowledge and consent, and what is her duty towards her mother in this case?

If the mother is capable of taking care of herself, or if there is someone to look after her if she is incapacitated, then there is no objection to the daughter traveling with her husband. The daughter must take care of her mother and fulfill her rights, and spend on her if she is wealthy and the mother is poor.