Circuits involving arrays of Josephson junctions have emerged as a new platform for exploring and simulating complex bosonic systems. Motivated by this advance, we develop and theoretically analyze a one-dimensional bosonic system with sublattice symmetry, a bosonic Su-Schrieffer-Heeger model. The system features electrostatically controlled topological mid-gap states that we call soliton states. These modes can be measured using either spectroscopy through a normal lead or admittance measurements. We develop a protocol to adiabatically shuttle the position of these topological soliton states using local electrostatic gates. We demonstrate a nearly perfect fidelity of soliton shuttling for timescales within experimental reach.