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So this came up in a recent game, when my opponent tried to tank shock to break my gaunts (who were unfortunately out of synapse at the time), but his raider would've went off the board. We weren't sure if the raider was supposed to stop or go off the board so we rolled-off and he won, but I'm just thinking, was it supposed to stop?

Yes.
Models are not allowed to move off the board other than when they are falling back.

5th question in the main FAQ.

Durn, ok thanks!