Yes. They may move through a friendly sea zone that contains an enemy-controlled Convoy Center or Route during noncombat movment, but they may not take control of it when doing so.
Sub Questions
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Never had this happen before suprisingly. J1 attack on the phillipines, the US sub survived and submerged. With the battle over, can the transports still unload their troops?
Second question, can an attacking sub instantly submerge after moving regardless of the presence of an enemy destroyer?
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Yes to both.
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Thanks!