Is it permissible to transfer an interest-based debt to a non-interest based debt or to a 0% interest rate debt, so the interest can be reduced or eliminated? If permissible, can I do it for one of my family members using my card?

When someone is already in debt with an interest-bearing debt and that person can transfer the interest-bearing debt to a 0% interest or to a reduced interest rate debt, then it becomes permissible to transfer the interest-bearing debt to a lower interest rate debt or to a 0% interest debt to avoid paying more interest.
It is permissible to do that only if you guarantee that you or your relative who is using your card to do the transfer to your account, will commit to paying it off before the grace period ends. If the grace period ends and that person has not fulfilled his commitment, you become obligated to pay it off and you can not allow the interest to incur on your account claiming that it was his fault for not committing to his promise to pay the debt before the grace period ends.