The primary nanostructure formation kinetics in a set of Fe-Cu-Nb-Si-B ribbons has been studied by the use of differential scanning calorimetry in both continuous-heating and isothermal measuring regimes. The nanostructure formation consists of two independent kinetic modes, namely the Johnson-Mehl-Avrami nucleation-and-growth mode and the mode which has been well described by the normal-grain-growth kinetic law. The nucleation-and-growth of the bcc alpha-Fe based phase is manifested as a rule in the early stages of the transformation. The normal-grain-growth-like mode is the major and principal kinetic characteristics of all FINEMETs independently of the content of Cu and Si. It is controlled by a specific rearrangement of niobium. After the start of the main nanostructure transformation, a long-range reordering of Si occurs during longer times in the FINEMETs containing Si. Fe75.6Cu1Si14B9.4 is a conventional metallic glass. It does not contain niobium, does not exhibit the normal-grain-growth-like kinetics and does not form the nanocrystalline (FINEMET-type) structure.