We have engineered the band gap profile of transmon qubits by combining oxygen-doped A1 for tunnel junction electrodes and clean A1 as quasiparticle traps to investigate energy relaxation due to quasiparticle tunneling. The relaxation time T-1 of the qubits is shown to be insensitive to this band gap engineering. Operating at relatively low-E-J/E-C makes the transmon transition frequency distinctly dependent on the charge parity, allowing us to detect the quasiparticles tunneling across the qubit junction. Quasiparticle kinetics have been studied by monitoring the frequency switching due to even-odd parity change in real time. It shows the switching time is faster than 10 mu s, indicating quasiparticle-induced relaxation has to be reduced to achieve T-1 much longer than 100 mu s.