Can the non-striker be run out at my end?

Yes, during play. If the batters are running and the stumps at your end are broken by a fielder before the batter has grounded their bat or body behind the popping crease, they're run out (on appeal). That decision is yours. Stand where you can see both the crease and the stumps as the throw comes in.

Before the ball is bowled, never: in Stage 1, if the bowler runs out a non-striker who has left their crease early, it's not out, however many times it happens. Use it to teach the batter to stay in their crease until the ball is bowled.