Here's some dense law for you -- a general rule of California law is that a prohibitive injunction is not stayed during the pendency of an appeal. Rubin v. American Sportsmen Television Equity Society, Inc. (1951) 102 Cal.App.2d 288, 290. A "prohibitive" injunction is one that forbids the defendant from doing something, contrasted with a "mandatory" injunction which compels the defendant to do something.
To be sure, a clever lawyer can phrase any mandatory command in the form of a prohibition (e.g., "Defendants shall not fail to deliver the disputed property to the plaintiff,") so there is an issue about whether the effect of an injunction is prohibitory or mandatory. The Perry case could go either way, as I see it. The order in Perry v. Schwarzenegger is at least debatably prohibitive, in that the judge has forbidden the defendants from enforcing or implementing Prop. 8. That effectively returns the law back to the way it was after the Marriage Cases were decided but before Prop. 8 won by a margin of about 600,000 votes in November of 2008. The argument that it is mandatory would be that it effectively incorporates the Marriage Cases, which did contain a mandatory component requiring that the trial court issue orders that would compel county registrars to issue marriage licenses to same-sex couples.
Now, the Rubin case decides an issue of substantive California law and not a rule of Federal procedure. Federal Rule of Appellate Procedure 8 is silent on the issue of what orders are or are not stayable while an appeal is pending. I've not had time to do the research to see if there is a cognate rule to Rubin in the Federal system. But Rubin is interesting because it purports to articulate a substantive rule of law rather than a procedural one.
If the Ninth Circuit agrees that Rubin is substantive rather than procedural, then the Ninth Circuit is required under the Erie Doctrine to follow it and demand a showing of extraordinary and irreparable harm to the appellants if the prohibitory order of Judge Walker is not stayed pending appeal by way of an application for a writ of supersedeas. Given that there is a substantial question as to whether the proponents -- the only ones whose appeal has even a chance of being recognized by the Ninth Circuit in the first place -- have standing at all, this would have to be some damn good lawyering by the proponents' counsel.