Hypervelocity stars (HVS) move so fast that they are not bound to the Galaxy. When they were first discovered in 2005, dynamical ejection from the supermassive black hole (SMBH) in the Galactic centre (GC) was suggested as their origin. The two dozen HVSs known today are young massive B stars, mostly of 3–4 solar masses. Recently, 20 HVS candidates of low mass were discovered in the Segue G and K dwarf sample, but none of them originates from the GC. We embarked on a kinematic analysis of the Segue HVS candidate sample using the full 6D phase space information based on new proper motion measurements. Their orbital properties can then be derived by tracing back their trajectories in different mass models of our Galaxy. We present the results for 14 candidate HVSs for which proper motion measurements were possible. Significantly lower proper motions than found in the previous study were derived. Considering three different Galactic mass models, we found that all stars are bound to the Galaxy. We confirm that the stars do not originate from the GC. The distribution of their proper motions and radial velocities is consistent with predictions for runaway stars ejected from the Galactic disk by the binary supernova mechanism. However, their kinematics are also consistent with old disk membership. Moreover, most stars have rather low metallicities and strong α-element enrichment, as is typical for thick disk and halo stars, whereas the metallicity of the three most metal-rich stars might indicate that they are runaway stars from the thin disk. One star shows halo kinematics.