An assembly includes a longitudinal rail oriented substantially parallel to a closing plane, in which the a movable furniture door is arranged in the closing position, and a transverse rail arranged transversely to the longitudinal rail. The transverse rail can be arranged substantially parallel to a depth direction of a side space of a stationary furniture part. A fastening device can fasten the longitudinal rail and/or transverse rail to the stationary furniture part. The fastening device has an adjustment device for adjusting the longitudinal rail and/or transverse rail to be substantially parallel to a longitudinal direction of the transverse rail relative to the stationary furniture part.