Moving the goalposts is responding to rebuttal or criticism by changing established definitions or declaring exceptions to the rule.
This is different to bringing up new evidence or backing to counter counter-arguments—moving the goalposts fundamentally changes what would have been a valid counter-argument in the first place.
For example, is Jeff Bezos an astronaut?
“The FAA's new policy order, which went into effect on July 20 — the same day the billionaire blasted into space (coincidence?) — states that in order for a person to qualify for the FAA Commercial Space Astronaut Wings Program, a person must meet a new specific set of requirements. First, they must reach an altitude higher than 50 miles above the surface of the Earth during flight, and they must also participate in activities during the mission that were ‘essential to public safety, or contributed to human space flight safety.’”
(Source: Salon)
The article accuses the FAA of changing the definition in order to prevent Jeff Bezos from qualifying. That would be moving the goalposts.
It wouldn't be moving the goalposts if the FAA simply reviewed and updated their definition of astronaut, and then checked afterwards whether Jeff Bezos fit the new definition. But in this case, the timing is certainly suspicious.