Round 1:
Korra has waterbending only: Korra should win very high dif, due to power and physicals.
Round 2:
Korra has earthbending and firebending: Katara should win high dif. Korra's a skilled firebender and earthbender, but Katara's better, and earth and fire don't really synchronize well.
Round 3:
Korra has earthbending and airbending: Katara should win very high dif. Earth and air synchronize well, but they're still Korra's two weakest elements, and I think that Katara should win more often than not.
Round 4:
Korra has firebending and airbending: Korra extreme dif or tie. We've already seen that Korra has zero issues evaporating large quantities of water while moving at high speeds in her fight against Unalaq, but despite Korra's offense and mobility advantage, Katara's good enough that she can't be discounted, and I'm not sure who'd win. It's worth noting that Katara also has very good feats against firebenders.
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