Inspired by the recent interest in the Sachdev-Ye-Kitaev (SYK) model we study a class of multi-flavored one- and two-band fermion systems with no bare dispersion. In contrast to the previous work on the SYK model that would routinely assume spatial locality, thus unequivocally arriving at the so-called 'locally-critical' scenario, we seek to attain a spatially-dispersing 'globally-SYK' behavior. To that end, a variety of the Lorentz-(non)invariant space-and/or-time dependent algebraically decaying interaction functions is considered and some of the thermodynamic and transport properties of such systems are discussed.