We study a discrete Schrödinger equation with arbitrary long-range hopping terms under the influence of an external force. The impact of long-range hoppings on the single-particle Bloch dynamics in the lattice is investigated. A closed expression for the propagator is given, based on which we analyze the dynamics of initially Gaussian wave packets. Our findings capture the anharmonic oscillations recently observed in zigzag lattices and furthermore provide a detailed quantitative description of the crossover between center-of-mass Bloch oscillations for wide wave packets and left-right symmetric width oscillations for narrow single-site excitations. The analytical results are shown to be in agreement with numerical simulations. A helix lattice setup for ultracold atoms is proposed where such hopping terms to far neighbors can be experimentally tuned to sizable values.