I will add my 2 cents by describing how current flows in a perfect inductor. This might help to understand how inrush current occurs in a real inductor.
Even in a perfect inductor, startup conditions can cause current flow in excess of the expected 'steady state' current flow. (More correctly, with a 'perfect' inductor steady state transients will persist; there is no unique 'steady state'.)
An inductor is simply a conductor arranged to interact with the magnetic field created when current flows in it. Every wire is somewhat an inductor. Inductors are usually set up as coils of wire, concentrating the magnetic flux and letting the same 'loop' of flux interact multiple times with the conductor.
In an inductor, current flow creates a magnetic flux. This magnetic flux stores energy. As with any circuit element, we can relate the current flowing through the inductor, the voltage applied to the inductor, and the energy stored in the inductor.
For a perfect inductor, the applied voltage sets the rate change of the magnetic flux coupled to the conductor. If you were to apply 1V to an inductor then the flux coupled to the conductor would be fixed at 1 Weber per second. Note that this doesn't say anything about the current flow! The applied voltage sets the rate of flux change.
Of course, since it is the current which creates the flux in the first place, the applied voltage is also indirectly setting the rate of current change for a given inductor. Each inductor will have a proportionality factor, called its inductance, which is the measure that relates current flow to flux coupled to the conductor.
In a 1 Henry inductor, 1 amp of current results in 1 Weber of flux coupled to the conductor. Apply 1V to a 1H inductor, and the coupled flux will change at 1Wb per second and current will change at a rate of 1A per second. Apply 1V to a 1uH inductor, and the coupled flux will still change at 1Wb per second, but now the _current_ changes at 1000000A per second.
What happens when you apply an AC voltage to a perfect inductor?
During the positive half of the AC cycle, total flux will become more positive (for a suitable sign convention). The flux will keep increasing for the entire positive half of the cycle. Similarly, during the negative half of the AC cycle the total flux will become more negative. The voltage and the duration of the AC half cycle determine the _change_ in flux experienced during that half cycle. The higher the voltage, the more the flux change. The longer the time the more the flux change.
Note that the above never tells us what the actual flux is; it simply tells us how the flux changes. It is necessary to know what the flux is at a given point in time to figure out what the flux will be later.
We can relate this to 'inrush' because it is the initial conditions of the inductor and the applied AC voltage that determine how the flux in and current through the inductor evolve over time.
So, for example, we could start with an inductor that has no flux in it (and no current flow), and apply an AC voltage. Depending upon where in the AC cycle we initially apply the voltage, we can figure out what the flux will do.
Say we take this inductor, with 0A flowing in it and zero flux, and apply AC right at the zero crossing going into the positive half cycle. The flux (and current) will increase to a maximum, and this maximum will be right at the end of the positive half cycle. Then on the negative half cycle the current (and flux) will drop back to zero. In this case, for a 'perfect' inductor, flux (and current) will cycle from 0 to + maximum and back to 0 again.
If instead we take this inductor and apply the AC voltage right at the peak of the positive half cycle, the flux (and current) will increase until the zero crossing...but because we have only carried half of a half cycle the peak flux (and current) will only be half that of the first example case. In the negative half cycle, the flux (and current) will drop from the positive maximum to zero and then go to the negative maximum, right at end of the negative half cycle.
In a nutshell: the peak flux coupling a 'perfect' inductor can vary by as much as 2x depending on where in the AC cycle an AC voltage is applied.
I am not going to continue the discussion to cover real inductor, but will add a point to consider: the applied voltage sets the _flux_ coupled by the inductor, not the current. If that flux is being carried by a core which can saturate, then this 2x variability in peak flux will make the difference between the core saturating or not.
-Jon