Three single crystals of Ni(Co)-Mn(Fe)-Ga(Cu) high-temperature magnetic shape memory alloys exhibiting a tetragonal non-modulated martensite have been produced. Extensive measurements of the key parameters responsible for the magnetic field induced twin boundary motion, such as magnetocrystalline anisotropy, twinning stress, magnetostress and tetragonality ratio have been carried out as a function of the composition and temperature. Universal relations between the indicated parameters have been disclosed. The feasibility of high temperature magnetic actuation is supported in, at least, one of the crystals, indicating that these materials are very attractive for high temperature applications.