I believe this issue could be resolved several ways, but first there are a few things to consider. Assuming the PC is the one initiation dimension door, or some variant that allows a teleportation like effect, as a free action and still retains the ability to act via Dimensional Agility or some other feat/ability/class feature there are a few scenarios here. If the PC and the enemy targeted both "jumped" at the same time via the same ability (keeping in mind that the enemy would need to be willing for Dimension Door...albeit there may be some other means of teleporting one's self and an enemy simultaneously against the enemy's will) and, baring some other feat/ability/class feature/skill (such as fly or a monk's slow fall) which may slow falling speed, would both be falling at the same speed and therefore within reach of an attack by the PC. If falling under Ge, or Earth like gravity, they would be falling at a rate of 9.8 m/s^2 or 32.2 feet per second squared. That's 6.44 vertical squares of downward movement per second. Since the Dimension Door/Teleportation effect was initiated as a free action, the PC and enemy would both be falling for the full six seconds. That's 193.2 feet of downward movement in a single round (without slow fall or some other condition acting upon one or both of the fighting pair)! The distance traveled here would be very important to keep in mind given that, unless the dimension door movement is beyond 195 feet straight up, both of the falling characters would impact some time before the six seconds of time elapsed for a full attack or full round of action.
Furthermore, each attack would create a force driving the combatants away from each other due to the lack of bracing (as CatLord alluded to when speaking about penalties to hit). Thus any character hoping to dimension door up, say 500 feet, and take a full attack action on another enemy that is also falling would need to control his/her momentum (via fly/acrobatics/etc.) each time he/she attacked to drive his/her self back toward the defender. This would be incredibly difficult to do while landing attacks. However, assuming it could be done, the defender would not retain their full ability to dodge because of the same bracing issue. You could consider allowing the defending character to retain a certain amount of dodge AC based upon an equivalent fly/acrobatic/etc.(perform dance perhaps?) check. Just keep in mind neither character's feet are planted so any ability to move is dependent upon the character's ability to manipulate body momentum and air flow.
Keeping those two things in mind (rate of decent and the ability to manipulate one's momentum after each attack) let's consider what would happen if the defender is able to fly (via wings/supernatural ability/spell effect/etc.) but was not before the dimension door. The defender may or may not be able to initiate flying until their next action (Technically the six seconds of a round occurs for all pcs and npcs with a bit of an overlap...pcs and enemies complete actions before each other successively in game as dictated by initiative. Thus, the first actor's actions within that six seconds theoretically takes place prior to another character's actions even though the six seconds of a round is the same for everyone...for game mechanic purposes we just state that each person has six seconds) This whole cause effect aspect of the six second round as it is played out would entail that a flying npc might not be able to initiate actually flying until their next action as they have already used their actions for the round. Personally, I would not GM this way, treating the ability to fly as more of an instantaneous reaction dependent upon the defending character's maneuverability in the air.
So, let's assume a scenario where the defender can fly and does so as an immediate action after being carried willingly with the Dimension Dooring individual, or a scenario where the defender was flying before the Dimension Door and the PC simply used Dimension Door to reach the defender. Baring the attackers ability to flow fall or fly him/her self, the attacker would fall 6.44 squares of vertical distance in a single second. Thus, the amount of time they are adjacent to their foe is going to be dependent upon the size of their enemy. So, assuming the attacker was smart enough to position the exit point of their dimension door with them slightly above their fellow traveling enemy or at the uppermost vertical square adjacent to their already flying enemy so that in either instance they the attacker would fall past their enemy (keep in mind they would be rolling acrobatics/fly checks to avoid attacks of opportunity as well as maintain the directional momentum to land blows and probably be taking severe penalties on those rolls...although positive fly checks would probably augment acrobatics checks and vice versa) they would have enough time to complete as many attacks as the amount of attacks they typically complete in a similar amount of time.
IE: If the targeted flying enemy is large this allows for four squares of vertical movement from the highest diagonal adjacent square above the targeted enemy to the lowest diagonal square of space just below the targeted enemy. The computation for figuring any characters rate of attack per square fallen would be X/6/6.44 (where X=the amount of attacks the PC/NPC has per round, 6 is the number of seconds in the round and 6.44 equals the amount of squares covered during free fall...the last may be altered via fly checks or slow fall, etc.) Jal'Tareth the 16th level fighter has two weapon fighting, improved two-weapon fighting, and greater two-weapon fighting giving him seven attacks per round and .181 attacks per square covered. Now, I would say that attack one takes place in square one. Thus, the second attack may take place when the rate of attack per distance fallen for our character Jal with seven attacks per six seconds adds up to 1 or more while still falling past the creature.
Using the formula mentioned above, let's plug in Jal's attacks per round to come up with 7/6/6.44 (Jal has failed his check to slow his decent so he is falling at the normal rate of decent). This comes out to .181 attacks per square fallen. Not counting the first square (as that is the location of the first attack executed) our PC would need to fall six additional squares before being able to attack again...so...the flying creature would need to threaten seven squares of vertical space at least for Jal to get two attacks off. Jal would also need to make fly/acrobatics a check after landing the first blow to see if he could stay close enough to make the second attack.
Using the same computation method above, a character with eight attacks per round could make a second attack after having fallen a total of six squares (one attack in the first square...a second after an additional five squares at a rate of .207 attacks per square). Thus, the enemy would need to threaten six squares of vertical space. Theoretically, if a character could make 16 attacks per round (multiweapon craziness as an example perhaps) that character's attack rate per square of fallen space would be .414 per square of fallen space. That character could make his/her first attack in square 1 and then fall 3 squares before being eligible to make a second and an additional 2 after that to be eligible for a third. Thus our 16 attacks per round character could get off two attacks on a large flying creature while free falling past it or three against a creature threatening six squares of space.
However, as mentioned much earlier, it is important to remember that just because a character is fast enough with his/her weapons to make the attacks this doesn't mean that his/her momentum will allow him/her to do so. Each check is the RAW equivalent of a split second decision/reaction to position one's body to counteract the opposing force created by a blow while free falling. Thus, fly/acrobatic checks would need to be taken into account to keep the character positioning his/her body mid-air after each attack in order to be in the right position to even make the attack (I would say a DC 20 at least) If the check failed I would say that the character had moved horizontally the equivalent of a five foot step away from the enemy (thereby provoking an AoO) because of the momentum of the attack and the failure to position his/her body in such away as to get them closer for additional attacks. This would hold true for ranged attackers as well (although moving 5 feet away at a time may or may not cause a ranged attacker issues depending upon how far away they were to begin with).
Also, simply falling past a flying enemy provokes attacks of opportunity (although one would need to take into account how many AoOs the defender got if any beyond the first based on the defenders attack rate...to be fair) and I would think that the only way acrobatics or mobility or even dodge to AC for the falling attacker would come into account at all against the AoOs provoked would be via a fly check of at least DC 20 first and a second acrobatics/perform dance/some equivalent check of +15 (if not twenty) compared to the normal amount for moving through a square without provoking AoOs. So, this kind of Nightcrawler-ish action could be pulled off with some insane stats/amount of attacks per round/or ability to slow decent perhaps.
However, should the PC survive the attempt it may be a wonderful teaching moment. A free action Dimension Door with Dimensional Agility is only as good as a good pair of Boots of Levetation. ;)
I know that was crazy long...I've been working on the mechanics for GMing a world of floating islands with airships, etc. Hope this helped someone out there in the aether.