The search for MD4/MD5 collisions involves a lot of different concepts. To keep things simple, a differential path is chosen and messages are found so that the conditions put on the internal state values for the differential path to lead to a collision are satisfied.
The tunnels are transformations on the message that do not impact the status of these conditions up to some step $p_v-1$ of the algorithm, where $p_v$ is called point of verification: if they are fulfilled for the message M, they are also fulfilled for the transformed message.
Now tunnels are nice but one also need to find the proper messages to apply the tunnels to. Leurent's technique is meant to find colliding messages with a particular shape and therefore has some additional constraints. Hence he builds messages unorthodoxly (compared to the previous works such as Klima's) so that they meet the conditions only up to step $p_c-1$ of the algorithm where he calls a point of choice the value $p_c$. Between step $p_c$ and $p_v-1$ probabilities are at work and some freedom in the messages built so far is used to wait for the remaining conditions to be fulfilled.