The only constitutional issue with moving national elections is the 4th clause of Article 2:
The Congress may determine the Time of chusing [sic] the Electors, and the Day on which they shall give their Votes; which Day shall be the same throughout the United States.
Read literally, the only time that's fixed is the day that the electors cast their votes. In practice, this is generally interpreted to mean that Congress sets the election day, but it has to be the same day in the whole country. It is fully within Congress' power to change the date at their whim, but that gets into a question of practicality. States plan for their elections for months in advance - suddenly changing it close to the election day can throw all those plans into chaos.
On the state level, the Constitution (Article I, Section 4, Clause 1) says
The Times, Places and Manner of holding Elections for Senators and Representatives, shall be prescribed in each State by the Legislature thereof; but the Congress may at any time by Law make or alter such Regulations, except as to the Places of chusing [sic] Senators.
Congress has used the italicized power to define the national presidential voting day as the same one for the general election of Senators & Representatives. However, it leaves the manner up to the states, some of which have said "You have X amount of time to vote ahead of time, and we stop accepting votes and count it up on the official day".
Special elections to fill a vacancy are under different clauses than the above.
Article 1, Section 1, Clause 4
When vacancies happen in the Representation from any State, the Executive Authority thereof shall issue Writs of Election to fill such Vacancies.
Article 1, Section 2, Clause 2 (replaced by Amendment below)
... if Vacancies happen by Resignation, or otherwise, during the Recess of the Legislature of any State, the Executive thereof may make temporary Appointments until the next Meeting of the Legislature, which shall then fill such Vacancies.
When vacancies happen in the representation of any State in the Senate, the executive authority of such State shall issue writs of election to fill such vacancies: Provided, That the legislature of any State may empower the executive thereof to make temporary appointments until the people fill the vacancies by election as the legislature may direct.
The "Writs of election" mentioned here allow for the so called "Special elections" which are used to replace vacancies. These are up to the governor and/or legislature of each state to schedule, and they may choose to share them with the national election day or designate a separate day. They're also free to move them, subject to any state laws. However, the same logistical issues as with national changes apply here too.
tl;dr National elections can be changed by Congress. State elections can be changed by states. But there's logistical issues that would prevent it without extreme circumstances.