I am asking "how the initial turn-on transient decays into the steady
state where the voltage "over both arms" is equal to the battery
This question is likely best approached in the context of transmission line theory where we actually study such transients. In 'ordinary' (ideal) circuit theory, we use the lumped element model and make certain assumptions such that Kirchhoff's circuit laws hold (to good approximation).
But if you look at the turn-on transient at high enough resolution, you see that the battery initially doesn't 'see' the resistors at the 'far' end of the circuit but, rather, some characteristic impedance that is a function of the geometry of the circuit and connecting wires etc.
(The following will necessarily omit some finer points for the sake of clarity of exposition.)
That is, at turn-on, a 'wave' is launched from the battery 'down' the connecting wires towards the resistors. There is both a voltage wave and a current wave. At this point, the ratio of the voltage and current is determined by the characteristic impedance rather than the parallel connected resistors.
This wave propagates with speed $u \lt c$ and, upon reaching the first resistor, is (generally) partially reflected (back towards the battery) and partially transmitted (towards the second resistor).
The partially transmitted wave propagates to the second resistor where it is partially reflected (back towards the first resistor) and partially transmitted (towards a third resistor if it exists etc.)
The partially reflected wave from the second resistor propagates back to the first resistor where it is partially reflected (back towards the second resistor) and partially transmitted (towards the battery).
So, after a while, there is this superposition of many counter-propagating waves from the multiple transmissions and reflections. The voltage and current associated with each wave individually are related by the characteristic impedance however, the total voltage and current due to these is (quite remarkably) the steady state solution for the circuit.
If it helps, here's a more or less standard transmission line transient analysis to model high-speed switching response:
See how the source and load voltages are not the same initially but eventually converge as the multiple reflections superpose to the steady state solution.