If the card is obtained from Party A and can access accounts at a bank (any bank that is not Party A), the provisions of Regulation E, section 205.14(a) kick in (assuming there's no contract between Party A and the bank holding the account):
(a) Provider of electronic fund transfer service. A person that provides an electronic fund transfer service to a consumer but that does not hold the consumer's account is subject to all requirements of this part if the person:
(1) Issues a debit card (or other access device) that the consumer can use to access the consumer's account held by a financial institution; and
(2) Has no agreement with the account-holding institution regarding such access.
There are a number of changes to the compliance responsibilities incumbent on Party A, and the bank is "off the hook," except that it must cooperate with Party A in Party A's Section 205.11 error resolution efforts.
Assuming that there are no contractual agreements between your bank and the issuer of the card, and that the transactions arrive at your bank for posting via ACH, they should arrive as "POS" ACH entries, and you should have return rights for ODs, no-account, etc.
Whether Capitol One -- or anyone else filling the shoes of Party A -- would have to "eat" a return will be up to its contract with the cardholder. Presumably there will be some protections for the issuer.