In this situation, there are few things to take into consideration :
if you have a single booking reference for all the itinerary, then the airline considered that you could make it as they have allowed this booking and so you’ll be booked on a later flight if for some reasons you aren’t able to make it
if you have two separate bookings, then it will most of the time be your problem unless you can find an agreement with the airline but if you missed it for 10 minutes delay, it won’t be enough to justify a free rebook on a later flight with most companies
as mentioned in one of the comments, you should let the airline crew know about your situation. They’ll call some staff on the ground to escort you and make the connection happen.
I doubt that you’ll get a monetary compensation for being delayed in this situation. If the flight had 10 minutes delay, it isn’t enough to entitle you for a compensation, even if this generates further delay because of missed leg. The only thing you’ll eventually get is vouchers to drink something or a hotel night if the rebooked flight is the day after. The monetary compensation is usually related to flight delay, not to missed flight (ie when you aren’t able to take the flight).