Single-particle and collective microscopic dynamics of a classical Yukawa fluid are analysed on the basis of simple memory function and mode-coupling theories. Thermodynamic states for different screening regimes are correlated in terms of an effective hard-sphere packing fraction. The longitudinal current fluctuations and the velocity autocorrelation function are found to cross over continuously from the behaviour characteristic of classical fluids with short-range interactions to the dynamics of a one-component plasma as the screening parameter of the Yukawa potential is reduced. In particular, a simple mode-coupling analysis is capable of reproducing the increasingly oscillatory behaviour of the velocity autocorrelation function found in recent molecular dynamics simulations.