The Pickup and Delivery Problem (PDP) consists in defining a set of routes that satisfy transportation requests between a set of pickup points and a set of delivery points. This paper addresses a variant of the PDP where requests can change vehicle during their trip. The transshipment is made at specific locations called "transfer points". The corresponding problem is called the Pickup and Delivery with Transfers (PDPT). Solving the PDPT leads to new modeling and algorithmic difficulties. We propose new heuristics capable of efficiently inserting requests through transfer points. These heuristics are embedded into an Adaptive Large Neighborhood Search. We evaluate the method on generated instances and apply it to the transportation of people with disabilities. On these real-life instances we show that the introduction of transfer points generally brings non-negligible improvements (up to 9%). Experiments on related instances from the literature show clearly that the method is competitive with existing method.