Figure 1. Three-phase currents in a balanced system at various points in time. They always sum to zero.
If we could freeze frame the voltage waveform where L1 and L2 are positive maximum and L3 is negative, ...
That never happens. Have a look at Figure 1 points (1) and (2) and you will see that only one phase can be maximum at any time and at that time the other two are at half-maximum and in the opposite direction.
... what will the current direction through the windings look like?
They will look the same but with the minor complexity that the currents are out of phase with the phase-neutral voltages.
Will we at any point have the current from 2 different phases going towards each other?
Yes. At Figure 1 (2), 90°, the current in on the black phase is split in two leaving on the red and blue phases at half the input current. 60° earlier (1) blue and black are at 0.5 positive with red at peak negative.
The currents always sum to zero, even if the load is unbalanced. If you run the three wires through a clamp-on meter it will read zero. (This assumes that the is no neutral connection.)
From the comments:
At the point where phase 1 and 2 are at positive 0.5 and phase 3 is at -1, which direction will the current between a1 and b2 be facing, up or down?
Figure 2. In the situation described the current through the yellow winding will be zero.
Also the current from phase 2 on winding B will pass through the winding and head towards a1 from b2. Won't these currents clash?
If currents "clash" in the mathematical analysis then you add them taking care to observe the polarity or sign of the signal. That way the answer can be zero.