Heterostructures receive intensive attentions due to their excellent intrinsic properties and wide applications. Here, we investigate the natural physical properties and performances of strain BN-MS2 (M = Mo, W) heterostructure by density functional theory. Different to compressive monolayer MS2, corresponding BN-MS2 heterostructures keep direct band-gap characters because effects of charge transfer on anti-bonding dz2 orbitals are stronger than those of Poisson effect. Mexican-hat-like bands without magnetic moments are observed at strain BN-MS2 heterostructures when the compression is enough. Consequently, electron mobilities of strain BN-MS2 heterostructures are slightly reduced at first and then enlarged with increasing compressive strain. Note that, strain BN-MS2 heterostructures reduce the band edges of MS2 layers and extend their application in photocatalytic water splitting. But just the n-type and p-type Schottky barriers of devices with strain BN-MS2 heterostructures are reduced and even vanished with the increasing tensile and compressive, respectively. Besides, electron mobilities of strain BN-MoS2 and BN-WS2 heterostructures can be enhanced to 1290 and 1926 cm2 V -1 s-1, respectively, with increasing tensile strain. Interestingly, the exciton binding energies of strain BN-MS2 heterostructures exhibit oscillation variations, different to those of strain monolayer MS2.