After passing articles of impeachment the House of Representatives' role is not done. During the trial in the Senate the House collectively becomes "managers" of the trial and act as the plaintiff in a more traditional trial setting. After voting on the articles they are introduced in the Senate by the House of Representatives, not a Senator. This means that the Senate does not have control over the timeline for the original motion. In fact, they don't even get to control when the House presents the articles to the Senate (that is controlled by the Secretary of the Senate and happens automatically).
From Senate Impeachment Rules:
I. Whensoever the Senate shall receive notice from the House of
Representatives that managers are appointed on their part to conduct
an impeachment against any person, and are directed to carry articles
of impeachment to the Senate, the Secretary of the Senate shall
immediately inform the House of Representatives that the Senate is
ready to receive the managers for the purpose of exhibiting such
articles of impeachment agreeably to said notice.
II. When the managers of an impeachment shall be introduced at the bar
of the Senate, and shall signify that they are ready to exhibit
articles of impeachment against any person, the presiding officer of
the Senate shall direct the Sergeant-at-arms to make proclamation, who
shall, after making proclamation, repeat the following words, viz:
"All persons are commanded to keep silence, on pain of imprisonment,
while the House of Representatives is exhibiting to the Senate of the
United States articles of impeachment against ;" after which the
articles shall be exhibited, and then the presiding officer of the
Senate shall inform the managers that the Senate will take proper
order on the subject of the impeachment, of which due notice shall be
given to the House of Representatives.
III. Upon such articles being presented to the Senate, the Senate
shall, at 1 o'clock afternoon of the day (Sunday excepted) following
such presentation, or sooner if so ordered by the Senate, proceed to
the consideration of such articles, and shall continue in session from
day to day, (Sundays excepted) after the trial shall commence, (unless
otherwise ordered by the Senate,) until final judgment shall be
rendered, and so much longer as may, in its judgment, be needful.
Before proceeding to the consideration of the articles of impeachment,
the presiding officer shall administer the oath hereafter provided to
the members of the Senate then present, and to the other members of
the Senate as they shall appear, whose duty it shall be to take the
Once the motion is made in the Senate, the Chief Justice is automatically called upon and sworn in as the presiding officer of the impeachment trial. During the trial of Andrew Johnson articles were passed on February 29th in the House, introduced on March 4th in the Senate and the Chief Justice was sworn in March 5th. To give you an idea of the power wielded by the Chief Justice in such cases, Chief Justice Rehnquist was famously strict during President Clinton's impeachment trial, not even allowing Senators to leave the floor for the entire length of the trial once it began (to the frustration of many Senators).
So the trial would proceed in a timely manner once the House passed articles of impeachment. The articles of impeachment do not expire. "[T]he articles of impeachment against Judge Alcee Hastings were presented in the Senate during the second session of the 100th Congress [...] but were still pending trial by the Senate in the 101st Congress, when the House reappointed managers [...]. The articles of impeachment against President Clinton were presented to the Senate after the Senate had adjourned sine die for the 105th Congress, and the Senate conducted the trial in the 106th Congress."
Sec. 5 . Effect of Adjournment
An impeachment may proceed only when Congress is in session. 3 Hinds Sec. Sec. 2006, 2462. However, an impeachment proceeding does not expire with adjournment. An impeachment proceeding begun in the House in one Congress may be resumed in the next Congress. 3 Hinds Sec. 2321; 111-1, Jan. 13, 2009, p __. An official impeached by the House in one Congress may be tried by the Senate in the next Congress. Manual Sec. 620; 3 Hinds Sec. Sec. 2319, 2320.
Although impeachment proceedings may continue from one Congress to the next, the authority of the managers appointed by the House expires at the end of a Congress; and managers must be reappointed when a new Congress convenes. Manual Sec. 620. Managers on the part of the House are reappointed by resolution. Manual Sec. 604; Deschler Ch 14 Sec. 4.2. Thus, the articles of impeachment against Judge Alcee Hastings were presented in the Senate during the second session of the 100th Congress (100-2, Aug. 3, 1988, p 20223) but were still pending trial by the Senate in the 101st Congress, when the House reappointed managers (101-1, Jan. 3, 1989, p 84). The articles of impeachment against President Clinton were presented to the Senate after the Senate had adjourned sine die for the 105th Congress, and the Senate conducted the trial in the 106th Congress. Manual Sec. 620.