It was my understanding that they cannot arrest a congress critter when he is on his way to congressional business - hence the claim "I gotta go vote". This is not an unlimited immunity from arrest and prosecution.
Not exactly correct.
In the United States, Article One of the United States Constitution provides that members of Congress shall be immune from arrest in going to and departing from sessions and while Congress is in session except for cases of "Treason, Felony, and Breach of the Peace". A similar protection applies to members of the Parliament of the United Kingdom during a session of Parliament and 40 days to either side of a session. This protection also extends to Peers of the United Kingdom at all times, although it's only been used twice since 1945.
The first two are somewhat self-explanatory; it has been suggested that the third is deliberately somewhat vague. The doctrine thus established is called Congressional immunity; it arose out of the necessity to prevent a vengeful executive from arresting members of the legislative branch on a pretext to prevent them from taking actions that the executive might find to be displeasing. In recent years, this doctrine has been used to prevent members from being stopped and held for speeding on their way to sessions; this apparently is not a "breach of the peace", where as perhaps another misdemeanor such as "drunk and disorderly" might be construed to be such.
Most states of the United States and most other English-speaking jurisdictions have extended this privilege to members of their legislatures on the theory outlined above.