The transport minister works best with small transports, since it only deals with whole-loads of population.
What you probably want to do is micromanage your pop, and keep the high-growth planet just below max.
Then skim one turn's growth of population off the top, and send them to your major production worlds to slave away
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As for capturing population, the easiest way is probably to just take your big transports and a good sized fleet of direct-fire ships (no missiles!)
Fill one or two transports with troops, and save the rest for hauling away the slaves.
1) Be sure your transports have their strategy set to "capture planet"
2) Be sure your fleet is set to break formation.
3) Attack planets with lots of population.
The warships will blast the planet until the weapon platforms are gone, and then the troop transports will land.
A huge transport full of troops should capture almost any planet in stock and in most mods.
On the next turn, the planet should be yours, and you can load all your troops back up along with the people.