Generally speaking, you are correct. The broker will usually close at the first opportunity at or beyond the critical level. But beware the generalities! You need to check the T&Cs to make sure the default, the broker operating on a best efforts basis, applies to any account you might have with them.
It happens most frequently in equity markets, where companies report earnings after or before the close, generating a reaction on the open. Brokers closing out surpassed stops at the open is one of the reasons that the open can be so volatile for stocks in the news.
Of course, the process can work to investors' benefit, gapping beyond their take-profit levels as well as their stops.
But for obvious reasons, it's infinitely more problematic with stops, where disputes can and do end up in court. The mess following the SNB's abandonment of the EURCHF peg in Jan15 is the case in point here. With the banks physically unable to close out institutionally-sized positions en masse, what was the correct price point to work with here? And how could any two banks using different values from each other not end up in court? It took one of if not the biggest earthquake in FX history to generate this situation; but we do have an exercise in the "worst case" in recent market memory.
Only 7 months later, you had an equal and opposite disaster with the meltdown in US ETFs. Except the irony here was that the index imposed an automatic firebreak (put in place after the 2010 Flash Crash) on ~1,700 funds. On screen, the Powershares Low Volatility Portfolio was down 48% (you really couldn't make it up)... but nobody could get stopped out because of the halt to trading. When the halts rolled off, and prices snapped back, the gaps closed and many never actually saw their stops hit in open trading. Talk about an "inelegant" happy ending.
Of course, this is why many brokers offer a guaranteed stop, taking the gap risk for a fee. And it's precisely why so many institutional portfolios and hedge funds are willing to wear the time decay on put protection. Long gamma trumps any broker's best efforts.