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In an inductor, back emf is induced as there comes flux change. It is said that back emf decreases as time passes (when inductor is just connected to DC source).

When back emf decreases current increases in inductor. Current increases flux associated with inductor.

Back emf comes obeying Lenzs law. (Back emf is to fight flux change but why it is supporting flux change by decreasing...) Decreasing back emf is again increasing flux change. Then why and how back emf reduces?

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Then why and how back emf reduces?

The back emf reduces as the current increases, only because the inductor also has some parasitic resistance, and/or you've connected it in series with a resistor, so that some of the applied voltage is required to maintain the current through the resistance and isn't available to drive further increases in flux (with the associated back-emf).

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