What causes the phase to change in a VOR?
The latest decades VOR are Doppler VOR (DVOR) where the bearing signal is produced by a 14 m diameter circular array of fixed Alford loops antennas. An additional central antenna radiates the reference signal.

Thales Doppler VOR 432, source
DVOR is an improvement of the conventional VOR (CVOR) where the bearing signal is encoded by frequency modulation, more reliable than amplitude modulation used in the CVOR. FM is not produced in the station itself, it is a Doppler artifact created by the apparent displacement of the bearing signal source relatively to the receiver.
DVOR principle
The so-called dual sideband DVOR uses one pair of diametrically opposed antennas from the array at a time, switching to the next pair counter-clockwise at the angular speed of 30 revolutions per second (30 Hz, 1,800 rpm).

One antenna radiates an unmodulated signal at $\small f - 9.96$ kHz ($\small f$ being the frequency of the VOR)

The opposite one radiates at $\small f + 9.96$ kHz.


The receiver sees the two constant frequency sources moving along the circle, one approaching, one moving away. Because they move relatively to the receiver, their frequency is altered by a Doppler shift proportional to the relative velocity.

Doppler shift on the DVOR sidebands antennas
For the receiver, the observed signal frequency is increased for the approaching antenna and decreased for the retreating antenna. By chance for engineers, the Doppler shift applies in opposite quantity on the antennas, and the frequency remains mirrored relatively to the carrier frequency $\small f$ whatever the shift.
At the same time the central antenna radiates a carrier at frequency $\small f$, modulated in AM by the reference signal. This AM signal is not subject to the Doppler shift, as the signal source is fixed.
For the receiver the signal from the central antenna and the two signals from the rotating antennas appear as the three components of single HF signal of frequency $\small f$, modulated in amplitude at 30 Hz (by the reference) and a subcarrier at 9.96 kHz, itself modulated in frequency at 30 Hz (by the variable signal). This signal is therefore entirely compatible with one from a CVOR.
The Doppler shift depends only on the receiver relative bearing
While the timing of the reference is imposed by the VOR, the timing of the subcarrier modulation depends entirely on the aircraft position. E.g. the Doppler shift is null when the axis of the antenna pair is aligned with the aircraft direction, and both antennas move only from left to right from a receiver standpoint:

Doppler shift is null when the receiver is aligned with both antennas&lt;/sup
Similarly, and when the distance of the receiver is large compared to the array diameter, the maximum shift is observed when the antenna pair is perpendicular to the previous position (in practical the parallax error can be ignored).
Therefore the delay between the reference and the variable signal is representative of the bearing. Periodic signal phases are actually angles between 0 and 360°, so the bearing is determined at any time by subtracting the current phase of the FM variable signal from the current phase of the AM reference using a phase detector.

Phase of a sine signal
Note the receiver works only by comparison of the phases, their actual values is irrelevant, only the difference is meaningful. If that were not the case, the DVOR wouldn't be compatible with a CVOR receiver.
The CVOR also relies on the phase difference. It works by forming a rotating directional cardioid pattern by space modulation. The result of this modulation faces north when the variable signal which electronically steers the pattern has the value 135°. So the reference was aligned on this value and given the value of 135° for north (in the DVOR north is associated with the reference value 0°).
The diameter of the array gives the variable signal phase its correct value
The tangential velocity when moving along the perimeter of a circle is $\small v = \pi d \times rpm$. Applied to our array: $\small v = \pi \times 14 \times 30 \approx 1,320$. The relative tangential velocity of the sideband antennas is between -1,320 m/s and +1,320 m/s. The value of the Doppler shift for a given velocity is: $\small s = v \cdot f / c$. Applied to a frequency of 110 MHz, the Doppler shift is $\small 1.32e3 \times 1.1e8 / 3e11 = 484$, that is between about -484 Hz and +484 Hz. The exact diameter of the array is actually a bit less, in order to obtain the FM swing of the conventional VOR (480 Hz). This way the DVOR signal can be received by a CVOR receiver (there is no specific DVOR receiver).
As explained earlier, this shift is not encoded by the VOR into the variable signal but added by the Doppler effect in space. However the receiver sees the result and interprets it has if it had been generated by the VOR.
When the never modulated FM subcarrier is nevertheless processed by the receiver demodulator, it produces a 30 Hz sine which phase varies according to the Doppler shift which produced it, therefore depending on the correct diameter of the array.
VOR frequency spectrum
The receiver sees:

A signal at frequency $\small f$, AM modulated by the 30 Hz reference

A signal at $\small f - 9.96$ kHz, FM modulated by a 30 Hz sine (1,800 rpm).

A signal at $\small f + 9.96$ kHz, FM modulated by a 30 Hz sine but with the opposite shift.


It just decides those three signals are components of a single one from a single antenna: The first signal is the carrier, and the two other are sidebands.
It composites the whole thing this way:

This is a carrier of frequency $\small f$, modulated in AM by a 30 Hz signal (reference),

This carrier is also AM modulated by a subcarrier at 9.96 kHz.

The subcarrier is itself FM modulated by a 30 Hz signal (variable) with a modulation swing of 480 Hz.



VOR signal spectrum