When do constraints, like ParentConstraint and AimConstraint, evaluate?

I frequently employ the LateUpdate method on MonoBehaviours to locate or modify animated Transforms, after their animations have been evaluated. This works with both Timeline and Mecanim state machines, and has any number of uses; such as temporarily pinning props to character's hands, or offsetting the rotation of a character's bones.

I have recently taken to using the AnimationModule's new Constraints, such as ParentConstraint and AimConstraint. They're fantastic, but I have just discovered that they do not evaluate in time for their output to be used in LateUpdate. If I query the position of a constrained Transform in LateUpdate, it appears to return its position from the previous frame, not unlike querying an animated Transform's position in Update.

The effect is fairly catastrophic, in that the position of any Transform, with a Constraint anywhere above it in the hierarchy, is more or less impossible to find. Even using WaitForEndOfFrame to retrieve the value inside a Coroutine seems to yield an incorrect result.

When do Constraints actually evaluate, and is there any way to get the current or final position of a Transform being driven by a constraint? Constraints have been such a delight to work with otherwise, I would hate to have to stop using them. Any help would be much appreciated.

As an example of why one might need to know the position of a constrained Transform; I have a Spline system where a number of Transforms can serve as the vertices of a Bezier Curve. The resulting Spline can then be sampled at intervals, and its samples can be used to position other GameObjects, or the vertices of a LineRenderer, or whatever. We use it for cables, hoses, ropes and wires. This system works marvelously with normal animations, but once constraints are involved - such as constraining the nozzle of a hose to a character's hand - there's no safe method in which to sample the Bezier Spline, and the resulting samples become a jittery mess.