When the Senate immigration bill was voted on in the Judiciary committee, Sen. Cornyn voted against it
The senior Texas senator opposed the bipartisan immigration plan when it cleared the Judiciary Committee on a 13-5 vote last month. http://trailblazersblog.dallasnews.com/2013/06/sen-john-cornyn-on-immigration-deal-not-a-picasso-that-cant-be-touched-pushes-tighter-border-measures.html/
When the Senate voted to invoke cloture and move to a final vote (end the filibuster) on the bill, Sen. Cornyn voted against the motion
Earlier today, both Texans also voted against bringing the bill to a final vote. The votes were no surprise. Both openly opposed the bill. http://trailblazersblog.dallasnews.com/2013/06/cornyn-cruz-vote-no-on-senate-immigration-reform-bill.html/
And when the Senate immigration bill came up for final passage on the Senate floor, he voted against it
Texas Republican Sens. John Cornyn and Ted Cruz both voted against the Senate immigration reform bill that passed this afternoon, 68-32. http://trailblazersblog.dallasnews.com/2013/06/cornyn-cruz-vote-no-on-senate-immigration-reform-bill.html/
After voting against the legislation in Judiciary committee, this is what he said at the time on the procedural vote: Cornyn voted to proceed on the bill so he could offer his amendment."The bottom line is that if border and national security cannot be guaranteed in this bill, I cannot and will not support it," Cornyn said. http://www.chron.com/news/houston-texas/houston/article/hed-4594893.php
Your account of procedural votes is correct, but that’s not what Stockman is claiming. He is deceptively implying Sen. Cornyn voted for the bill:
“You see, Cornyn wants millions of illegal aliens to register to vote because Karl Rove tells him they’ll vote Republican…And as I told you earlier, Cornyn helped Democrats pass amnesty in the Senate last year. Well, I automatically killed that bill in the House with a “blue slip” resolution.”…Cornyn and Obama are working to ram amnesty down your throat.
Voting for a procedural motion so you can try to dramatically change a bill you oppose is a far cry from voting for the final legislation that moved to the House. Each time there was a vote on the substance and policy of the full legislation, he voted against it.