Culturally responsive therapy is a stance more than a branded protocol, so the strongest evidence is not for a treatment with that exact name. What has been studied is cultural adaptation of psychological interventions, and the link between a therapist's multicultural competence and how people actually do in treatment. That is related empirical work, and it is the honest place to look. It is not a trial of a single manual called culturally responsive therapy, and you should read the findings that way.
Hall, G. C., Ibaraki, A. Y., & Huang, E. R. (2016), Behavior Therapy. A meta-analysis of 78 studies found culturally adapted psychological interventions were associated with better psychopathology outcomes than other conditions, including a medium effect when compared with unadapted versions of the same intervention.
Soto, A., Smith, T. B., & Griner, D. (2018), Journal of Clinical Psychology. Two meta-analyses found culturally adapted interventions were associated with better outcomes on average, and that client-rated therapist cultural competence was strongly associated with treatment outcome, while therapists' self-ratings of competence were not.
Benish, S. G., Quintana, S., & Wampold, B. E. (2011), Journal of Counseling Psychology. A direct-comparison meta-analysis found culturally adapted psychotherapy was associated with better outcomes than unadapted, bona fide therapies for people from culturally diverse populations. This paper compares adapted versus unadapted treatment. It does not test a standalone culturally responsive brand.