You can’t simply transfer your own benefit, but the person looking after your children should be able to get some additional help from Work and Income. It will depend on their current financial situation.
You might need to help the caregiver by giving them a letter (or meeting other Work and Income requirements) to explain that they are now looking after the child.
If you will have your child with you in a Mothers with Babies Unit, you may be able to receive a family assistance benefit.
For more information, see “Pregnancy and childcare in prison“