See here. Details to come. UPDATE 1: The opinion is here.
From my initial post on the cert grant: “I expect that the Supreme Court will answer the question presented in the negative, ruling that a state may set up a system whereby nonmembers of unions must affirmatively consent before the union can deduct money for political purposes. I was surprised by the original Washington state supreme court ruling holding that such a system violated the union’s first amendment rights, in that it was not as narrowly tailored as an ‘opt-out’ provision (whereby monies for political purposes are automatically deducted unless the nonmember asks that they not be deducted).”
UPDATE 2: Before oral argument in Davenport, a number of reporters asked me what impact this case was likely to have on how the Court is likely to decide the WRTL case. I didn’t think it would have much impact, and now that the Court has issued the opinion, I continue to believe that Davenport seems unlikely to have any impact on how the Court decides WRTL. The union in Davenport tried to use Austin and Bellotti to make an argument about its right to spend unlimited sums in ballot propositions. Speaking for 6 members of the Court (Chief Justice Roberts, and Justices Alito and Breyer would not have reached the issue), Justice Scalia found the argument irrelevant:

The Supreme Court of Washington’s description of s 760 notwithstanding, our campaign finance cases are not on point….The cases on which respondent relies deal with governmental restrictions on how a regulated entity may spend money that has come into its possession without the assistance of governmental coercion of its employees. See, e.g., Bellotti, Austin. As applied to public-sector unions, s 760 is not fairly described as a restriction on how the union can spend ‘its’ money; it is a condition placed upon the union’s extraordinary state entitlement to acquire and spend other people’s money.

(original emphasis)
Marty Lederman says Davenport’s logic should mean the reaffirmation of Austin in WRTL. Regardless of what logic would appear to dictate, I see the Davenport opinion (which remember is signed by Justices at both poles on the Autin question—Scalia and Thomas at one pole and Stevens, Souter and Ginsburg at the other) as saying rather explicitly that this case has no relevance to the Austin question.
UPDATE 3: Brad Smith responds to Marty.