How to perform builk update within mysql stored procedure
Assuming that _batch_request_detail_ids = "1,2,3", try this :

CREATE DEFINER=`root`@`%` PROCEDURE `Update_Batch_Request_Detail_Bulk`(
IN _batch_request_detail_ids VARCHAR(5000),
IN _processing_status INT
)
BEGIN
     /**************** Set statement added***********/
    SET @sql =concat("UPDATE batch_requests_detail
    SET processing_status = ",_processing_status," 
    WHERE batch_requests_detail_id  IN (",_batch_request_detail_ids,");");
    PREPARE stmt1 FROM @sql;
    EXECUTE stmt1;
    END$$

DELIMITER ;