I don’t know why, but Mr. Switch never seems to give any references. I, personally, find giving page numbers allows others to open the book and find all the relevant text to the situation. I’m not beyond failure, no one is.
Transports: Page 31 from the instructions that come in the book
A transport can load cargo before, during and after its move. A transport can pick up cargo, move 1 sea zone, pick up more cargo, move 1 more sea zone and offload the cargo at the end of its movement. It may also remain at sea with the cargo still aboard. A transport can offload into a hostile territory only during an amphibious assault.
Please do note, that transports may not load units from a hostile zone. That means you cannot attack SZ 55 and load 2 infantry from W. Canada onto your transport in SZ 55.
Page 25 states:
Rockets: If you have the Rockets development, your antiaircraft guns
can act as rocket launchers. You can attack enemy production with your rockets during the
opening fire step of combat. This is the only situation in which antiaircraft guns can attack. From
each territory, one rocket may launch an attack. It attacks an industrial complex within 3 spaces.
Any industrial complex may be attacked by only one rocket launcher in a turn. The maximum
combined damage inflicted in one turn by all strategic bombing raids on one industrial complex
and any rocket attack on the same industrial complex is the territory’s income value. The
opponent must surrender that many IPCs to the bank.
Now that does not say anything about moving the gun. However:
In this phase, you can move any of
your units that did not move in the
combat move phase or participate
in combat during your turn. This
is a good time to gather your units,
either to strengthen vulnerable
territories or to reinforce units at
the front. You cannot move into
hostile territories in this phase.
You will notice that the rules clearly show that you may not move any unit that participated in combat during your turn. Opening fire is part of the Combat Phase of your round, thus, the AA Gun participated in Combat. (Page 21 of the box rules.)
As for moving AA Guns after capturing them:
Page 25 continues:
Usable by Invading Forces: If a territory is captured, any antiaircraft guns there are also captured. They
can be used by the capturing player in future combats. Antiaircraft guns are never destroyed, except when
a transport carrying one is sunk. If you move an antiaircraft gun into a friendly territory, place one of your
control markers under it. If you liberate a territory containing a captured antiaircraft gun, control reverts to
the original owner.
Now this does not SPECIFICALLY state that you may not move the AA Gun after capturing it. However, it is commonly accepted that AA Guns that were captured were involved in the battle, even if no aircraft were shot at, and thus, cannot move per the rule quoted above from page 21.
If that justification is not sufficient, you can always fall back on the justification that the AA Gun cannot move just after being captured for the same reason fighters and bombers cannot land in just captured territories.