ESC is essentially a pwm controlled pair of mosfets per motor winding, one mosfet shorting battery minus to one motor winding end, the other mosfet shorting battery pluse to the other motor winding end. Therefore the amplitudes of winding current and battery current are equal.
No, they’re not. Pick any log if you want proof, or read motor theory.
Power in = power out (+losses)
Power in=Vin x Iin
Power out=Vout * Iout
Now, Vin is not the same as Vout as motors are not driven in short circuit, and as you correctly say the voltage out is regulated with pwm switching to get an effectively lower voltage. To achieve energy balance the current therefore is higher on the motor side. This difference tapers off with higher rpm as more and more of the dc voltage is applied to the motor side to achieve the requested rpm and current.
I understand what you are saying.
I am talking about what is happening per pulse from electronics point of view.
Mosfets short battery to a winding during pulse duration t. Therefore winding accepts all the battery voltage for this duration. Because winding is an inductor, current starts at zero and rises linearly to some value at the end of the pulse (but not exceeding maximum esc current).
I wish i could find a shorter and easier explanation just like that but it’s too late 
Motor current is higher than battery current up to nearly full rpm.
I see, so it acts like a step down. Motor is more than an inductor. Ok so I will have to read more about motors 
. Thanks
Yes, it’s also more complicated than this since there is bemf from the motor.
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very interesting, I try to take off with my boogie (dual motor flipsky 6384 / NonVesc FT85BS / battery 13S6P 70A max) I record the log on one side only, after 251s, full power, motor rise to 60A. after 2s the rpm is max, but not enough power to lift my 70kg on a 1200cm2 surf foil. I would be curious to see other logs, the current peak does not seem normal to me, and I wonder how the rotation speed is obtained.
Several things, rpm for you with this motor is erpm/7 so rpm is only 1800 and the measured power is also only 400w. There’s the issue. Seems like the propellers are losing traction, flexing or sucking air - if there’s no rpm limit. Try the noload speed, do you reach full (kv x voltage) rpm?