The self-diffusion coefficients (D) of the cation and anion in the ionic liquids 1-hexyl-3-methylimidazolium and 1-octyl-3-methylimidazolium hexafluorophosphates ([HMIM]PF6 and [OMIM]PF6) and 1-butyl-3-methylimidazolium and 1-octyl-3-methylimidazolium tetrafluoroborates ([BMIM]BF4) and ([OMIM]BF4) have been determined together with the electrical conductivities (kappa) of [HMIM]PF6 and [BMIM]BF4 under high pressure. The pressure effect on the transport coefficients is discussed in terms of velocity cross-correlation coefficients (VCCs or fij), the Nernst-Einstein equation (ionic diffusivity-conductivity), and the fractional form of the Stokes-Einstein relation (viscosity-conductivity and viscosity-diffusivity). The (mass-fixed frame of reference) VCCs for the cation-cation, anion-anion, and cation-anion pairs are all negative and strongly pressure dependent, increasing (becoming less negative) with increasing pressure. VCCs are the more positive for the stronger ion-velocity correlations; therefore, f+ - is least negative in each case. In general, f- - is less negative than f+ +, indicating a smaller correlation of velocities of distinct cations than that for distinct anions. However, for [OMIM]PF6, the like-ion fii are very similar to one another. Plots of the VCCs for a given ion-ion correlation against fluidity (reciprocal viscosity) show the fij to be strongly correlated with the viscosity as either temperature or pressure are varied, that is, fij approximately fij(eta). The Nernst-Einstein deviation parameter, Delta, is nearly constant for each salt under the conditions examined. It is emphasized that nonzero values of Delta are not necessarily due to ion pairing but result from differences between the like-ion and unlike-ion VCCs, because Delta is proportional to (f+ + + f- - - 2 f+ -). The diffusion and molar conductivity (Lambda) data are found to fit fractional forms of the Stokes-Einstein relationship, (LambdaT) proportional, variant (T/eta)(t) and Di proportional, variant (T/eta)(t), with t=(0.90+/-0.05) for all these ionic liquids, independent of both temperature and pressure within the ranges studied.