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We have performed ab initio molecular dynamics simulations to study the alkali-metal dynamics in the Al-doped (KAl0.33W1.67O6 and RbAl0.33W1.67O6) and undoped (KW2O6 and RbW2O6) defect pyrochlore tungstates. The K atoms exhibit novel rattling dynamics in both the doped and undoped tungstates while the Rb atoms do not. The KAl0.33W1.67O6 experimental thermal conductivity curve shows an unusual depression between ~50 K and ~250 K, coinciding with two crossovers in the K dynamics: the first at ~50 K, from oscillatory to diffusive, and the second at ~250 K, from diffusive back to oscillatory. We found that the low-temperature crossover is a result of the system transitioning below the activation energy of the diffusive dynamics, whereas the high-temperature crossover is driven by a complex reconstruction of the local potential around the K atoms due to the cage dynamics. This leads to a hardening of the K potential with increasing temperature. This unusual reconstruction of the potential may have important implications for the interpretation of finite-temperature dynamics based on zero-temperature potentials in similar materials. The key result is that the novel K rattling, involving local diffusion, leads to a significant reduction in the thermal conductivity. We suggest that this may open a new route in the phonon engineering of cage compounds for thermoelectric materials, where the rattlers are specifically selected to reduce the lattice thermal conductivity by the mechanism of local diffusion. [ABSTRACT FROM AUTHOR]