WASHINGTON—Senate Republicans moved to take their first step Thursday toward funding new spending on border security and the military, while Democrats tried to put GOP lawmakers on the record on uncomfortable issues at the start of President Trump’s new term.

Senators began chugging through an all-night budget “vote-a-rama” Thursday, a crucial, if dreaded, step toward unleashing a $340 billion package President Donald Trump's team says it needs for mass deportations and security measures that top the Republican agenda.

The measure conflicts with a House budget resolution that Trump has endorsed, but Republican senators see their version as a fallback plan.