Reinforcing works thusly:
Load one transport that is on your home coast with fresh units. Move that transport to meet up with the next nearest ships in the chain. Select the units and click the Load Units button and choose one of the other ships that have movement. Even though your land units are out of movement, they can still be transferred to the other ships that still have movement points! Now move the newly loaded ship to meet up with the next in the chain and repeat. The last step is to move the last ship in the chain into the captured city on the new continent, select the units and click Unload button. They will be inside the city now. This all happens in 1 turn. Next turn, move the ships back into position and repeat with the other set of ships that you didn't use this time.
By leaving 2 ships at each link in the chain, you can reinforce every turn. This reduces overseas warring time multiplicitively long before airports. It even works with galleys in classical era! It also allows you to go heavy on siege and city raiders in your initial attack and bring over city garrisons and siege reinforcements very quickly afterward. The AI will never have a chance
Thanks for your detailed explanation!
One important point worth noting is that when you transfer land Units from one ship to another, make sure to Shift + click on your land Units, so that they can Load from one ship to another as a group. It may sound obvious once you're used to doing it, but I know of at least one player who was incredibly frustrated with the process of transfering land Units from ship to ship because he was trying to do it one land Unit at a time, until after hearing that point. If you use the right-side Shift key on most keyboards, which is close to the letter L key, you can press "l" (the lower-case letter "L") once you have selected all of your land Units to transfer from one ship to another, saving a bit on mouse movements.
Also, if you don't have 2 ships at each link in the chain, you can still send land Units every turn if you are able to get a newly-built ship arriving at the starting point of the chain on every turn. On the following turn, the ship that arrived at the first link in the chain will have all of its movement points and it can sail land Units to the ship which was the first link in the chain on the last turn, which in turn can sail land Units to the shift was the first link in the chain two turns ago.
If you ever have any downtime where you don't have land Units to send over during a turn, then on a square where you have 2 ships, you can sail 1 ship backward in the chain. Say, at your arrival point, you have 2 ships--you can send one of them back. Then, the second last link of the chain has two ships and you can send the one with movement points backward in the chain--and repeat this process all of the way back so that you end up with 2 ships at the start of the chain. Assuming that for the next several turns, you'll have enough land Units arriving to be able to send over one group of land Units per turn, then on one of those turns, you won't need to have a newly-built ship arriving at the start of the chain, since you'll have already brought an extra ship back to the start of the chain.
"But, I don't have a City which can produce 1 ship per turn! So, I can't get a newly-built ship to the start of the chain on every turn!"
Well, you could build a Drydock to increase your Hammer output, if you've learned Steel.
But, the more practical answer is that you don't have to do so--just have multiple Cities building new ships for the start of the chain. It may take a City 3 turns to build a ship and 3 turns for that ship to arrive at the start of the chain, then 1 more turn for that ship to have its full movement points on the square at the start of the chain, but with multiple Cities building such ships, you can keep a steady flow of those ships arriving. Assuming that you can produce sufficient land Units (or that you have a large enough queue of waiting land Units) to keep your chain of ships busy each turn, then with a bit of planning, your multiple ship-building Cities can have one or more new ships arriving at the start of the chain on each turn.
It is okay if a newly-built ship arrives at the start of the chain earlier than needed (i.e. on the same turn as another newly-built ship), which means that on a later turn, you can afford not to have a newly-built ship arriving at the start of the chain, since said ship arrived on an earlier turn and is "ready and waiting."
If you ever do get two complete chains of ships, due to having the last ships in the chain start sailing backward on the turn after they have unloaded their cargo at the destination and then continuing to move back in the chain on future turns (do so on your turn after you've shuttled land Units over for the turn, so that you don't make mistakes of which ships to send backward and so that if you do make a mistake, you might have an extra ship in place with movement points which can make up for your mistake), then as long as you can have 2 new ships arriving at the start of the chain on each turn, you can actually send over two full groups of land Units per turn.
Extending this idea, if you are able to over-produce ships and the end of your chain has already started looping backward, you can send extra newly-built ships forward in the chain, thereby getting two complete sets of ships in the chain before the first ship has had a chance to loop all the way back to the start of the chain. Then, you just need enough land Units (6 for Galleons) that need shipping over each turn plus 2 new ships arriving at the start of your chain on each turn and you'll have two full sets of Units shipping over each turn, rather than just one set.
