The clash between Republicans and unions that caught fire in Wisconsin last week escalated Monday: Labor leaders planned to take their protests to dozens of other capitals and Democrats in a second state considered a walkout to stall bills that would limit union power.

The protests have ignited a wider national debate over the role of labor unions and who should shoulder sacrifices as states scramble to tackle yawning budget deficits. Governors in both political parties are looking for union concessions as they struggle to...