Each photo in an image burst can be considered a sample of a complex 3D scene: the product of parallax, diffuse and specular materials, scene motion, and illuminant variation. While decomposing all of these effects from a stack of misaligned images is a highly ill-conditioned task, the conventional align-and-merge burst pipeline takes the other extreme: blending them into a single image. In this work, we propose a versatile intermediate representation that consists of a two-layer alpha-composited image plus flow model constructed with neural spline fields -- networks trained to map input coordinates to spline control points. Our method is able to, during test-time optimization, jointly fuse a burst image capture into one high-resolution reconstruction and decompose it into transmission and obstruction layers. Then, by discarding the obstruction layer, we can perform a range of tasks including seeing through occlusions, reflection suppression, and shadow removal. We validate the method on complex synthetic and in-the-wild captures and find that our method, with no post-processing steps or learned priors, outperforms existing single-image and multi-view obstruction removal approaches.