The studied problem concerns the non-emergency transportation of patients. Some disabled or old patients cannot go back and forth to the hospital by their own to get cares. In this context, transport companies are involved to provide medical transportation of patients.The originality of our problem is to consider equity on the ``long'' term (typically one month), with requests dynamically revealed. Compared to standard equity models (that apply to a single period), it gives the opportunity to accept unbalanced routes at some periods, as long as equity is preserved on the whole horizon. We investigate different solution approaches and evaluate computationally how addressing equity this way allows limiting its impact on routing costs.