The mounted combat rules are weak to begin with, and you are going beyond them. This would be a GM call and require house rules to achieve the concept you desire.
As the rules are, you wouldn't have any action to force a movement on your opponent after the charge, so you couldn't move them. Additionally, there are no rules for a snag net being tied to another creature (or yourself for that matter) rather than being held in hand.
I personally could see allowing the mount to make a Drag combat manuever check instead of making an attack as part of the charge. This would reposition, but not damage or trip the attacker. It would not give you a free trip attack. If you wanted to trip them, you would have to grab the trailing rope on your next turn and attempt a trip maneuver as normal.
Generally forced movement doesn't in Pathfinder doesn't cause damage in Pathfinder. If I was feeling generous, I might allow something similar to what Gust of Wind does with tiny creatures, making it 1d4 non-lethal damage per 10' that were dragged (after they were prone).
I ran into this once when my party was having a chase scene on a horse-drawn carriage and they were being chased by guards on horseback. One of the PC's fell off but managed to grab a cargo strap and was being dragged behind the carriage.
We couldn't find any rules for it, so we made a rule on the fly and agreed that they would take 1d6 falling damage for every 30ft traveled while being dragged behind the carriage.
Your situation seems like you would like to use a mechanic like this on a regular basis though. I think you're going to have to get with your GM and create some house rules.
That’s kind of what I was figuring. Ok, we will make up some rules. Thanks!
Some House Rules to consider:
You should come up with appropriate DC's to break free of your net while dragging an enemy, which should probably have a higher DC than normally escaping a net. For reference, escaping a net is a DC of 20 Escape Artist check. But while moving, I would consider an increase of +2 DC, plus an additional +1 DC for every 30ft you're traveling in a round, with an Escape Artist check each subsequent round to become free. For example, a Horse traveling at 50ft would have a DC of 23, or a horse traveling at 100ft (charging) would have a DC of 25.
In order to move at full speed or faster in a round while dragging someone behind in a net requires a Ride check (or a Strength check from the mount), otherwise you could only move at half speed. Additionally consider the Drag rules on this for a basis of creating this house rule.
Your dragged opponent in your net takes 1d6 falling damage for every 30ft traveled. Save for half(?) with a Reflex save if the Escape Artist check fails.
While being dragged, retrieving items from your backpack is impossible. Retrieving items from readily available pouches cannot be quick drawn. If you have the Quick Draw feat, retrieving readily available item or a weapon requires a move action. If you do not have the Quick Draw feat, retrieving an item or weapon is a full-round action.
While being dragged, casting spells with somatic components is a DC 15 plus Double the spell's level, or the spell fails. Casting verbal spells is a 15 plus spell's level, or the spell fails.