The mobility of individual adatoms and of different structural elements in the O/Cu system is analyzed on the basis of scanning tunneling microscopy data. A value of D almost-equal-to 10(-14) cm2 s-1 for the coefficient of self-diffusion at 300 K on the bare Cu(110) surface is derived from the motion of steps. Supply of Cu adatoms by "evaporation" from steps rather than the migration of individual O or Cu adatoms on the terraces is rate limiting for the "added row" reconstruction processes, leading to a slowdown of the transformation with increasing conversion. The kinetics for the (1 x 1) --> (2 x 1) reconstruction is shown to follow that behavior, supporting a supply-controlled rather than a growth-controlled mechanism for (2 x 1) formation. The role of lateral interactions between neighboring-Cu-O-Cu-strings and of lateral displacements of entire such strings for the stabilization and growth of (2 x 1) islands and for the removal of antiphase domain boundaries (Ostwald ripening) are discussed. Conversion into the higher coverage c(6 x 2)O phase, again associated with a change in topmost layer Cu density and thus with mass transport, occurs in the direct vicinity of the location of the activated process, contrary to (2 x 1)O formation, and can be described as a local solid-solid transformation.