I am familiar with what happened in the Senate, but not the house. The Senate passed a resolution to expel ten southern senators. Some has seen terms expire, some resigned. Some did abandon seats.

The Senate Historical Office has the most concise explanation:

Sixty-four years later, at the start of the Civil War, senators again turned to this constitutional safeguard. Between December 1860 and June 1861, 11 of the nation's 34 states had voted to withdraw from the Union. What was the status of their 22 senators at the beginning of the 37th Congress? Some were no longer senators because their terms had expired. Others sent a letter of resignation. Still others, believing their seats no longer existed, simply left without formal notice. Several remained, despite their states' departure.

During a brief special session in March 1861, weeks before the start of hostilities, the Senate decided to consider these seats as vacant to avoid officially recognizing that it was possible for a state to leave the Union.

On the Fourth of July 1861, with open warfare in progress, President Abraham Lincoln convened Congress to deal with the emergency. With all hope of reconciliation gone, the Senate took up a resolution of expulsion against its 10 missing members. The resolution's supporters argued that the 10 were guilty, like Blount years before, of conspiracy against the government. In futile opposition, several senators contended that the departed southerners were merely following the dictates of their states and were not guilty of personal misconduct.

On July 11, 1861, the Senate quickly passed Senate Daniel Clark's resolution, expelling all 10 southern senators by a vote of 32 to 10. By the following February, the Senate expelled another four senators for offering aid to the Confederacy. Since 1862, despite considering expulsion in an additional 16 instances, the Senate has removed no member under this provision.