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Russia and China don’t really have carrier battle groups, so they don’t use submarines to protect them.
The whole Soviet naval strategy for war with the US was to block reinforcement of NATO forces in Europe by cutting off the sea lanes from the continental US. Since the US Navy (and the NATO allies) had vastly superior surface forces, the Soviets relied on submarines. As such, many Russian submarine designs were built specifically as anti-surface platforms and carrier killers, with extremely high speeds (the one-off *Papa*-class submarine could go 45 knots–over 50 miles per hour–and the *Alfas* were capable of over 40kts as well) and powerful, long-range weapons like very long range torpedoes and anti-ship missiles.
Diesel subs are slower because they don’t have sufficient battery capacity to propel them at high speed for any length of time, though modern air-independent propulsion technologies can help alleviate this.
In any case, the problem with high speed in a submarine is that they make a lot of noise, which means they’re easier to detect. Ideally an enemy would position submarines in the likely avenues of approach for carrier group, convoy, or whatever the target is, and wait. When the target shows up, engage it and run away. Rinse and repeat.