Because the class is about private litigation to enforce civil and constitutional rights, our discussion of legislative immunity focuses more on how it limits accountability for violations of individual rights rather than the important structural role it plays in allowing the legislature to check an overweaning executive.
A few examples in the news.
First (H/T: Noah), Rep. Ro Khanna took to the House floor to identify six people whose names had been redacted in the Epstein.
Second, a grand jury declined to indict the six Democratic members of Congress who posted a public video reminding service members of their obligation to disobey unlawful orders. This comes alongside Sen. Mark Kelly's lawsuit (Kelly is one of the six and a retired Navy officer) to enjoin an administrative proceeding to reduce his rank and pension. All six will, if it comes to it, raise Speech-or-Debate immunity as a defense to any attempt to punish or sanction them for that video. Of course, this gets tricky because immunity does not extend to public-facing statements. Many believe Hutchinson v. Proxmire and Gravel (to the extent they were trying to publish the Pentagon Papers in a book) were wrong, that public statements constitute an essential part of the legislative function.
Third, a whistleblower filed a complaint against Director of National Intelligence Tulsi Gabbard that made its way to the relevant committees and sparked some under-seal back-and-forth. Of course, any member of the House or Senate could read the whistleblower complaint (or the details of Gabbard's responses) on the floor.
That the first and third do not happen very often owes to the norms of the body (which are important) and the power of each house to punish its members--the important check on abuse of legislative immunity (wherever the power to punish may come from).