  Sinusoidal Bloch oscillations appear in band structures exposed to external
fields. Landau-Zener (LZ) tunneling between different bands is usually a
counteracting effect limiting Bloch oscillations. Here we consider a flat band
network with two dispersive and one flat bands, e.g. for ultracold atoms and
optical waveguide networks. Using external synthetic gauge and gravitational
fields we obtain a perturbed yet gapless band structure with almost flat parts.
The resulting Bloch oscillations consist of two parts - a fast scan through the
nonflat part of the dispersion structure, and an almost complete halt for
substantial time when the atomic wave packet is trapped in the original flat
band part of the unperturbed spectrum, made possible due to LZ tunneling.
