When someone is tied down, it means to be restricted by responsibilities.
To restrict someone or something in movement.
Constrain, confine, or limit.
1. Our new baby has really tied us down, we haven’t had date night in forever!
2. I’d like to travel more, but I’m tied down with my job.
3. He finds that a desk job ties him down.
4. Ms. Redd is tied down at the moment, can I take a message?
bind, confine, constrain, curb, curtail, deter, govern, hamper, hinder, keep down, prevent, repress, restrict, subdue, suppress