Ternary solid solution of (0.94-x) Bi0.5Na0.5TiO3-0.06BaTiO3-xBiAlO3 (BNT-BT-BA) lead-free piezoceramics was prepared using solid state reaction. Effect of BiAlO3 addition to Bi0.5Na0.5TiO3–BaTiO3 ceramics on structure, phase stability, strain behavior, dielectric, ferroelectric, pyroelectric, and piezoelectric properties was systematically investigated and a schematic phase diagram was proposed based on the phase transition temperature. The correlation between ferroelectric-to-relaxor transition temperature TF-R and depolarization temperature Td was shown by the comparison with Td determined by thermally stimulated depolarization current measurements and TF-R obtained from dielectric results, and investigated by temperature-dependent the stability of electrically induced long-range ferroelectric order and piezoelectricity. Accompanied with the TF-R shifted down below room temperature as the substitution of BiAlO3, electric-field-induced long-range ferroelectric order was disrupted, giving rise to a large strain of 0.37% with normalized strain Smax/Emax = 533 pm/V at a critical composition x of 0.02. Temperature-dependent changes in polarization and strain hysteresis loops in combination with XRD measurements and Raman-spectra analysis suggested that the large strain response should be attributed to the electric-field-induced phase transformation between ergodic relaxor and ferroelectric phase.