Process vs. Measurement Noise in Kalman Filter Tuning

You are running a Kalman filter for state estimation on a mobile robot, and the filter's output is too sluggish: it lags real motion and is slow to react to genuine changes in the state. A colleague suggests "just increase $Q$" (the process noise covariance). Explain what $Q$ and $R$ (the measurement noise covariance) physically represent, how the ratio between them governs the steady-state Kalman gain, and what the concrete trade-offs are of increasing $Q$ versus decreasing $R$. Finally, describe how you would diagnose whether the filter is mistuned in a principled way rather than tuning by trial and error.