I would rule that your mount gets to make a Reposition check to move the enemy.
But like Frodo, if this were for a PFS game I'd say you need to release the grapple or stop.
Assuming the character survives you don't get to automatically move the enemy. I would allow your mount a CMB check vs the creature's CMD to pull the creature along.
There is a rule that you can move a grappled foe with you, but only at half speed.
Here it is:
MoveYou can move both yourself and your target up to half your speed. At the end of your movement, you can place your target in any square adjacent to you. If you attempt to place your foe in a hazardous location, such as in a wall of fire or over a pit, the target receives a free attempt to break your grapple with a +4 bonus.
You've omitted important sections of the rules for grapple.
Grapple
As a standard action, you can attempt to grapple a foe, hindering his combat options. If you do not have Improved Grapple, grab, or a similar ability, attempting to grapple a foe provokes an attack of opportunity from the target of your maneuver. Humanoid creatures without two free hands attempting to grapple a foe take a –4 penalty on the combat maneuver roll. If successful, both you and the target gain the grappled condition (see the Appendices). If you successfully grapple a creature that is not adjacent to you, move that creature to an adjacent open space (if no space is available, your grapple fails). Although both creatures have the grappled condition, you can, as the creature that initiated the grapple, release the grapple as a free action, removing the condition from both you and the target. If you do not release the grapple, you must continue to make a check each round, as a standard action, to maintain the hold. If your target does not break the grapple, you get a +5 circumstance bonus on grapple checks made against the same target in subsequent rounds. Once you are grappling an opponent, a successful check allows you to continue grappling the foe, and also allows you to perform one of the following actions (as part of the standard action spent to maintain the grapple).
Move: You can move both yourself and your target up to half your speed. At the end of your movement, you can place your target in any square adjacent to you. If you attempt to place your foe in a hazardous location, such as in a wall of fire or over a pit, the target receives a free attempt to break your grapple with a +4 bonus.
Damage: You can inflict damage to your target equal to your unarmed strike, a natural attack, or an attack made with armor spikes or a light or one-handed weapon. This damage can be either lethal or nonlethal.
Pin: You can give your opponent the pinned condition (see Conditions). Despite pinning your opponent, you still only have the grappled condition, but you lose your Dexterity bonus to AC.
Tie Up: If you have your target pinned, otherwise restrained, or unconscious, you can use rope to tie him up. This works like a pin effect, but the DC to escape the bonds is equal to 20 + your Combat Maneuver Bonus (instead of your CMD). The ropes do not need to make a check every round to maintain the pin. If you are grappling the target, you can attempt to tie him up in ropes, but doing so requires a combat maneuver check at a –10 penalty. If the DC to escape from these bindings is higher than 20 + the target's CMB, the target cannot escape from the bonds, even with a natural 20 on the check.
So, you are mounted and use hamatula's strike. You strike the enemy with a piercing weapon and make a grapple check. Success means you impale the enemy and both gain the grappled condition. Then the grapple rules come into play. You've already used your standard action to make an attack, and you don't get a full attack because you've moved on your mount. On your next turn you could make another grapple check and with success you could choose to move the enemy at half your movement speed. Of course, this movement is in place of dealing damage or the other available options.
What the OP is asking is if he gets to automatically move the creature as a result of grappling. The answer is no. Strict RAW would say he has to stop or release the weapon as a free action to continue moving. It also doesn't describe what kind of action it is to remove the impaled weapon from the enemy.