Democrats’ election overhaul is expected to clear a major Senate hurdle, but the road ahead is unclear.

A key Senate committee is expected to vote as soon as Tuesday to advance Democrats’ sweeping elections overhaul, stamping its approval on a landmark voting rights expansion as Washington tries to blunt ballot restrictions by Republican-controlled statehouses.

The debate and anticipated vote in the Senate Rules Committee are a significant milestone for liberals who have made the bill, H.R. 1, their top legislative priority. If enacted, it would effectively override laws emerging in states like Georgia and Florida that raise barriers to vote with national requirements — like automatic voter...