Animating an inherited widget from a child UserWidget class appears to work, but the bindings don't resolve properly and the animation is null at runtime. This workflow should either be supported, or the UMG animation editor shouldn't expose those bindings if it's not an acceptable workflow.
1. Create a UserWidget titled ParentWidget. Add a canvas panel and a border to it, and mark the IsVariable checkbox for the border.
2. Create a second UserWidget, ChildWidget, that inherits from ParentWidget. Do not add anything to it's WidgetTree so that it inherits the tree from it's parent
3. Create an animation on ChildWidget and click the add track button. You'll be able to select the inherited border via the All Named Widgets option. Animate the border
4. Add the widget to the screen and play the animation. Note that the animation is null and does not play.
Note: In 5.2, a warning appears when compiling this widget: UMG Animation 'ChildAnim' from 'ChildWidget_C' is trying to animate a non-existent widget through binding 'Border_0'. Please re-bind or delete this object from the animation.
There's no existing public thread on this issue, so head over to Questions & Answers just mention UE-169479 in the post.