The Senate late Wednesday moved toward adopting a Republican budget blueprint that would pave the way for a $70 billion increase for immigration enforcement and the eventual reopening of the Department of Homeland Security.

But first, senators waded through an overnight marathon of rapid-fire votes, known as a vote-a-rama, that put the two parties’ dueling political messages on vivid display six months before the midterm elections.

Republicans, who are using the budget plan to lay the groundwork to eventually push through a filibuster-proof bill providing a multiyear funding stream for President Trump’s immigration crackdown, used the so-called vote-a-rama to highlight their hard-line stance on border security, seeking to portray Democrats as unwilling to safeguard the country.

Democrats tried and failed to add a series of changes aimed at addressing cost-of-living issues such as health care, food and energy prices, seizing the opportunity to hammer Republicans as out of touch with and unwilling to act on the concerns of everyday Americans.

Here’s what to know about the budget plan and the nocturnal ritual senators are engaging in before adopting it.

Republicans are seeking a way around a filibuster on D.H.S. funding.

The budget blueprint is a crucial piece of Republicans’ plan to fund the Department of Homeland Security and end a shutdown that has lasted for more than two months. After Democrats refused to fund immigration enforcement without new restrictions on agents’ tactics and conduct, the G.O.P. struck a deal with them to pass a spending bill that would fund everything but ICE and the Border Patrol. Republicans said they would fund those agencies through a special budget bill that Democrats could not block.

“We can fix this with Republican votes, and we will,” said Senator Lindsey Graham, Republican of South Carolina and the Budget Committee chairman. “Every Democrat has opposed money for the Border Patrol and ICE at a time of great peril.”

In resorting to a new budget blueprint, Republicans were laying the groundwork to deny Democrats a say in what would be in the bill. But they also submitted themselves to a vote-a-rama, in which any senator can propose unlimited changes to such a measure before it is adopted.

After the Senate passes the budget measure, it will go to the House, which must adopt it before lawmakers in both chambers can draft the legislation funding immigration enforcement.

Democrats used the moment to hammer Republicans on affordability.

Democrats took to the floor to criticize Republicans for supercharging funding for federal immigration enforcement rather than moving legislation that would address Americans’ concerns over affordability.

“This is what Republicans are fighting for,” said Senator Chuck Schumer, Democrat of New York and the Democratic leader. “To maintain two unchecked rogue agencies that are dreaded in all corners of this country instead of reducing your health care costs, your housing costs, your grocery costs, your gas costs.”

Democrats offered a host of amendments along these lines, all of which were defeated by Republicans — and that was the point. The proposals were meant to put the G.O.P. in a tough political spot, showcasing their opposition to helping Americans afford high living costs. Some G.O.P. senators crossed party lines to support them.

Republicans blocked proposals to address health care and grocery prices.

The G.O.P. thwarted an effort by Mr. Schumer to require that the budget measure lower out-of-pocket health care costs for Americans. Two Republicans who are up for re-election this year, Senators Susan Collins of Maine and Dan Sullivan of Alaska, voted with Democrats, but the proposal was still defeated.

Republicans also squelched a move by Senator Ben Ray Lujan, Democrat of New Mexico, to create a fund that would lower grocery costs and reverse cuts to food aid programs that Republicans enacted last year. Ms. Collins and Mr. Sullivan again joined Democrats.

Also defeated by the G.O.P.: a proposal by Senator John Hickenlooper, Democrat of Colorado, to address rising consumer prices brought by Mr. Trump’s tariffs and the war in Iran, and one by Senator Jon Ossoff, a Democrat who is up for re-election in Georgia, to require the budget measure to address health insurance companies denying or delaying access to care.

Republicans sought to amplify their hard-line immigration message.

While Republicans had fewer proposals for changes to their own budget plan, they also sought to offer measures that would underscore their aggressive stance on immigration enforcement and dare Democrats to vote against them.

Mr. Graham offered an amendment to allocate funds toward a deficit-neutral reserve fund relating to the apprehension and deportation of adult immigrants convicted of rape, murder, or sexual abuse of a minor after illegally entering the United States. It passed unanimously.