Placing robots outside controlled conditions requires versatile movement representations that allow robots to learn new tasks and adapt them to environmental changes. The introduction of obstacles or the placement of additional robots in the workspace, the modification of the joint range due to faults or range-of-motion constraints are typical cases where the adaptation capabilities play a key role for safely performing the robot's task. Probabilistic movement primitives (ProMPs) have been proposed for representing adaptable movement skills, which are modelled as Gaussian distributions over trajectories. These are analytically tractable and can be learned from a small number of demonstrations. However, both the original ProMP formulation and the subsequent approaches only provide solutions to specific movement adaptation problems, e.g., obstacle avoidance, and a generic, unifying, probabilistic approach to adaptation is missing. In this paper we develop a generic probabilistic framework for adapting ProMPs. We unify previous adaptation techniques, for example, various types of obstacle avoidance, via-points, mutual avoidance, in one single framework and combine them to solve complex robotic problems. Additionally, we derive novel adaptation techniques such as temporally unbound via-points and mutual avoidance. We formulate adaptation as a constrained optimisation problem where we minimise the Kullback-Leibler divergence between the adapted distribution and the distribution of the original primitive while we constrain the probability mass associated with undesired trajectories to be low. We demonstrate our approach on several adaptation problems on simulated planar robot arms and 7-DOF Franka-Emika robots in a dual robot arm setting.