Replies

Impeachment is an exception to the rule that “the matter dies.”
https://www.riddick.gpo.gov/UserData/SenateProcedures/Impeachment.pdf
11th page of that PDF, page 975 of Riddick’s Senate Procedure.

https://www.riddick.gpo.gov/UserData/SenateProcedures/Impeachment.pdf

At your cited source, Riddick's Senate Procedure, PDF page 11, document page 875.

Congress, Impeachment Carries Into New Congress: The Senate has carried an impeachment over into a subsequent Congress. Articles of impeachment against Harold Louderback, a United States district judge for the northern district of California were exhibited on March 3, 1933, at the end of the second session of the 72d Congress,¹⁰ and the trial occurred during the first session of the 73d Congress, concluding with acquittal on all articles on May 24, 1933.¹¹ At the end of the 100th Congress, the Senate adopted a resolution to continue into the 101st Congress the proceedings in the impeachment of Alcee L. Hastings, a United States judge for the southern district of Florida.¹²
10 Mar. 28, 1933, 72-2, Journal, p. 283-84. 11 May 24, 1911, 73-1, Journal, p. 344. 12 S. Res. 480, 100-2, Sept. 30, 1988
Consideration of Articles of Impeachment: On one occasion, just before 1:00 p.m. on the day following the day articles of impeachment were exhibited by the managers on the part of the House, unanimous consent was obtained that the Senate not proceed to the articles of impeachment until an unspecified future time.¹³
13 Aug. 7, 1986, 99-2, Record, p. 19779.

The Senate carried an impeachment trial over to a new Congress.

Hastings was impeached by the House during the second session of the 72nd Congress, and the trial occurred during the first session of the 73rd Congress.

The House cannot pass articles of impeachment and hold on to them for four years before delivering them to the Senate. Recall Nancy had to notify the Senate and deliver the Trump charges.

Once the House approves articles of impeachment, it must notify the Senate. Once the Senate is notified, it must take specified actions. It goes into session, remains in session, and does not consider business other than the impeachment.

It is only because the Senate is a continuing body that it may carry forward impeachment proceedings from one Congress to the next, as it did in the Alcee Hastings case.

See IMPEACHMENT, Selected Materials, Committee on the Judiciary, House of Representives, One Hundred Fifth Congress, November 1998,

105th Congress, 2d Session, Committee Print, Ser. No. 10, U.S. Government Printing Office, Washington, 1998

At book page 754, page 50 of Report of the National Commission on Judicial Discipline & Removal, August 1993.

Initiation of the Senate Trial
The Senate’s trial role commences only after receipt of articles of impeachment from the House at the bar of the Senate. The chairman of the House Managers demands that the Senate order the appearance of the accused to answer the charges, demands a conviction and appropriate judgment, and exhibits the articles. Unless a Rule XI committee is appointed, the Senate by resolution sets a date and time for proceeding to the consideration of the articles. The Senate’s jurisdiction is narrowly circumscribed by the case brought before it by the House. Absent articles, individual senators are, of course, free to refer complaints or evidence of judicial misconduct to the House of Representatives and to encourage an investigation by the appropriate committee.
The Senate has an obligation to render a final decision on articles of impeachment. It cannot take the matter under advisement indefinitely. Although the House of Representatives renews itself every two years, the Senate — as a continuing body — can carry forward an impeachment proceeding from one Congress to the next. In the proceeding involving Judge Hastings, the Senate did precisely that. This resulted in four Senators who had been Members of the House being excused from participation.

I would note that it is not apparent that an impeachment can not follow departure from office, although I have read online comments that it can't be done, it has been done.

[Senate Document 99-33]
[From the U.S. Government Printing Office]
99th Congress SENATE Document
2d Session 99-33

PROCEDURE AND GUIDELINES FOR
IMPEACHMENT TRIALS IN THE
UNITED STATES SENATE
(REVISED EDITION)
PREPARED PURSUANT TO SENATE RESOLUTION 439,
99TH CONGRESS, 2D SESSION

SUBMITTED BY SENATOR ROBERT C. BYRD AND SENATOR ROBERT DOLE
by
FLOYD M. RIDDICK, PARLIAMENTARIAN EMERITUS
OF THE UNITED STATES SENATE
and
ROBERT B. DOVE, PARLIAMENTARIAN OF THE
UNITED STATES SENATE

August 15, 1986.--Ordered to be printed
99th Congress Document
2d Session SENATE 99-33

At PDF page 61:

Resignation Does not Render Moot the Impeachment of the Respondent:
In the trial of William Belknap in 1876, Mr. Belknap resigned his office of Secretary of War and the question was raised
. . . whether W. W. Belknap, the respondent, is amenable to trial by impeachment for acts done as Secretary of War, notwithstanding his resignation of said office; . . .\210\
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\210\ May 4 1876, 44-1, Senate Journal, p. 928.
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The Senate resolved the issue by agreeing to the following resolution:
Resolved, That, in the opinion of the Senate, William W. Belknap, the respondent, is amenable to trial by impeachment for acts done as Secretary of War notwithstanding his resignation of said office before he was impeached.\211\
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\211\ May 29, 1876, 44-1, Senate Journal, p. 944.