Does artificial grass shift over time?

If so, how do I make sure that they stay in place?

Artificial grass won't shift over time, but it will expand and contract during temperature changes, just as everything else, the law of physics: when it is cold, materials contract, then it is hot, materials expand. Two things we can do to minimize the expansion and contraction are both at the installation stage:
First, you need to make sure to secure the edges of the artificial grass with nails every 5" apart;
Second, you need to add infills to provide extra weight on the grass, the general rule of thumb is to use 2 lbs of infills per square foot on average.

It wouldn't shift or move over time if it was properly installed. Make sure you choose your installers carefully and only trust the ones who's been in the business for a long time.

Just make sure that you have a solid base and surely, your artificial grass will not shift.