Graphene monolayers subjected to periodic lateral strains are artificial crystals that can support narrow bands with nontrivial valley topology. When the strain field varies slowly on the large superlattice scale, the low-energy physics in a single valley is described by the Dirac Hamiltonian with a valley-antisymmetric strain-induced pseudomagnetic vector potential. As a result of the superlattice potential, the energy spectrum fractures into a pair of doublet bands that connect at zero energy and a series of narrow singlet bands. The doublet bands have total Chern number $\mathcal{C} = \pm 1,$ and the singlet bands are Chern trivial. Because of particle-hole symmetry, the gaps carry half-quantized Hall conductance. This system hosts boundary spectra with an intrinsic polarity. These are analyzed by comparing the effects of periodic magnetic fields and strain-induced pseudomagnetic fields that respectively break and preserve time-reversal symmetry. In the former case, a Chern classification of the superlattice minibands with zero total magnetic flux enforces single counter-propagating modes traversing each bulk gap on opposite boundaries of a nanoribbon. For the pseudomagnetic field, pairs of counter-propagating modes migrate to the same boundary where they provide well-developed valley-helical transport channels on a single zigzag edge. We discuss possible schemes for implementing this situation and their experimental signatures. When subjected to a perpendicular electric field, the doublet bands in periodically-strained graphene are gapped out. The resulting bands carry significant Berry curvature and provide a favorable environment for the emergence of novel electron dynamics. In particular, the Berry curvature can induce an oscillating trajectory of an electron wavepacket transverse to an applied static electric field. Though analogous to Bloch oscillations, this novel oscillatory behavior is driven entirely by quantum geometry in momentum space instead of band dispersion. While the current from Bloch oscillations vanishes at large field strengths, the current from the geometric oscillations saturates to a nonzero plateau in the strong-field limit. In non-magnetic materials, the geometric oscillations are even under inversion of the applied field, whereas the Bloch oscillations are odd, a property that can be used to distinguish these two co-existing effects. We calculate these effects in periodically-strained graphene, and confirm that they can be experimentally measured at reasonable conditions.