D.C. Code § 11-2603
A person for whom counsel is appointed shall be represented at every stage of the proceedings from such person's initial appearance before the court through appeals, including ancillary matters appropriate to the proceedings. If at any time after the appointment of counsel the court finds that the person is financially able to obtain counsel or to make partial payment for the representation, it may terminate the appointment of counsel or authorize payment as provided in section 2606 of this chapter [11-2606], as the interests of justice may dictate. If at any stage of the proceedings, including an appeal, the court finds that the person is financially unable to pay counsel whom such person had retained, it may appoint counsel as provided in section 2602 [11-2602], and authorize payment as provided in section 2604 [11-2604], as the interests of justice may dictate. The court may, in the interest of justice, substitute one appointed counsel for another at any stage of the proceedings.
D.C. Code § 11-2603