Dielectric and mechanical spectroscopies have been used to investigate ferroelectric transitions and twin wall dynamics in the lead-free ceramic 0.5Ba(Zr$\mathrm{<mrow\; data-mjx-texclass="ORD">0.2</mrow>}$Ti$\mathrm{<mrow\; data-mjx-texclass="ORD">0.8</mrow>}$)O$\mathrm{<mrow\; data-mjx-texclass="ORD">3</mrow>}$−0.5(Ba$\mathrm{<mrow\; data-mjx-texclass="ORD">0.7</mrow>}$Ca$\mathrm{<mrow\; data-mjx-texclass="ORD">0.3</mrow>}$)TiO$\mathrm{<mrow\; data-mjx-texclass="ORD">3</mrow>}$ (abbreviated as BZT-50BCT), which is known to have a high piezoelectric coefficient (d$\mathrm{<mrow\; data-mjx-texclass="ORD">33</mrow>}$>545pC/N). Results from dynamical mechanical analysis in the frequency range 0.2-20 Hz and resonant ultrasound spectroscopy in the frequency range ∼0.1-1.2MHz confirm the existence of three phase transitions with falling temperature, at ∼360K (cubic-tetragonal), ∼304K (tetragonal-orthorhombic), and ∼273K (orthorhombic-rhombohedral). In comparison with BaTiO$\mathrm{<mrow\; data-mjx-texclass="ORD">3</mrow>}$, however, the transitions are marked by rounded rather than sharp minima in the shear modulus. The pattern of acoustic loss is also quite different from that shown by BaTiO$\mathrm{<mrow\; data-mjx-texclass="ORD">3</mrow>}$ in having a broad interval of high loss at low temperatures, consistent with a spectrum of relaxation times for interactions of ferroelastic twin walls. Differences in the dielectric properties also suggest more relaxor like characteristics for BZT-50BCT. It is proposed that the overall pattern of behavior is significantly influenced by strain heterogeneity at a local length scale in the perovskite structure due to the substitution of cations with different ionic radii. The existence of this strain heterogeneity and its influence on the elastic behavior near the transition points could be contributory factors to the development of adaptive nanoscale microstructures and enhanced piezoelectric properties.