Matching donor organs to patients in need is a difficult but important problem. A crucial factor in transplant outcomes is the cold ischemic time of the organ, which increases every time an organ offer is rejected. Despite this, acceptance dynamics have so far been neglected in favour of purely outcome driven offers. As a first alternative, we propose DynamITE, a novel causal organ allocation methodology that explicitly takes into account the acceptance behaviour over sequences of offers. DynamITE dynamically updates organ acceptance estimates, cold ischemic times (CIT) and causal effects throughout the matching process. We demonstrate that DynamITE improves early organ acceptance and maximizes patient life expectancy compared to current policies.