I may be off, but here are my two cents:
RAW states (my emphasis):
You create linked teleportation portals that remain open for the duration.
Choose two points on the ground that you can see, one point within 10 feet of you and one point within 500 feet of you. A circular portal, 10 feet in diameter, opens over each point. If the portal would open in the space occupied by a creature, the spell fails, and the casting is lost.
The portals are two-dimensional glowing rings filled with mist, hovering inches from the ground and perpendicular to it at the points you choose. A ring is visible only from one side (your choice), which is the side that functions as a portal. Any creature or object entering the portal exits from the other portal as if the two were adjacent to each other; passing through a portal from the nonportal side has no effect. The mist that fills each portal is opaque and blocks vision through it. On your turn, you can rotate the rings as a bonus action so that the active side faces in a different direction.
My interpretation of this is that, in order for the creature/object to exit on the opposite side, it must first COMPLETELY enter the entering side. I.e: you can't put an arm into it and grab something on the other side and drag it back. This is also supported by the statement about no effect in the opposite direction. Let's call this Non-partial-transfer scenario (NPTS)
As pointed out in the comments, there is also the possibility that GMJoe's interpretation is correct, and you can do partial-transfers where only you arm is on the other side of the portal, instantaneously. Let's call this Partial-transfer-scenario (PTS). I'll analyze things for both.
In the NPTS there is no way for your large vehicle to partially pass through: it is reasonable to assume that the part of the vehicle that is inside the portal exists in a ethereal/limbo state until the whole of the vehicle is inside the portal, and then it appears as a whole on the other side. You can still pull it out from the entrance before crossing completely, but it won't partially go through.
In both cases, the spell describes the aspect of the portals. There doesn't seem to be anything physical in them akin to, say, a wooden frame that the vehicle can get stuck to. Nor it seems the case that you can "run over" the portals (but more on this later)
So here is up to your DM. Here are the possibilities I envision:
The portal edges are unbreakable. A large object colliding with them at high speed will be damaged. If you envision them as sharp, the effect is that the vehicle will be cut in the shape of the portal: the part that goes through goes through, the parts cut out stay behind, otherwise it's just blunting damage and stopping the item in case it doesn't break.
The portal edges are breakable. A strong enough blow can destroy either portal, and this will dispel both sides. I'd model this as some sort of contest against the caster's DC.
The difference in case of items partially through is that, in the NPTS the part inside the portal is in a non-material limbo, while in the PTS they are just magically existing at the other side of the portal (possibly trampling things/people nearby the moment they spawn).
With regards to the "I close the gate while things are passing through": in the NPTS the "passing through" part is atomic, while in the PTS my arm is already on the other side.
You can model a dispel effect in different ways (possibly thematic to the dispel cause)
Option 1) Violent dispel: The occupying body is ejected violently, and takes dmg akin to a failed teleport (they smashed my portal!). In case of PTS, you can model this as an effective clean cut of the two parts (who doesn't love missing limbs?!), possibly mitigated by a successful DEX save (cat reflexes to pull in your arm)
Option 2) Gentle dispel: The occupying body is ejected without side effects, kinda like the floating down from levitate (they broke my concentration!). In this case, the PTS should probably eject you as a whole on either side.
In general, I'd eject the body on the entrance side. If your dispel cause is appropriate or you need it as a plot device, you can possibly envision a "decompression suction" kinda dispel that pulls you in before the dispel and pops you out on the exit side (useful if you want to split the party or to justify the PTS scenario).
Careful that PTS violent dispel is a VERY powerful way to do damage to a large creature chasing you through the portal and peeking its dragon head through where the barbarian smashes the portal (instakill unless it's something that spawns as sub-blobs?) .. so calibrate the possibility accordingly.