The $SWR$ is, in principle, defined as a ratio of maximum and minimum voltages that exist at different locations of the transmission line. These two different voltages $U_{max}$ and $U_{min}$ can be measured to determine $SWR$. - However, $SWR$ can also be determined by the absolute value of the reflection coefficient, $|\Gamma |$, according to
$$SWR = \frac{1+|\Gamma |}{1 – |\Gamma |} = \frac{U_{max}}{U_{min}} =
\frac{U_{forward}+U_{return}}{U_{forward}-U_{return}}$$
$$|\Gamma | = \sqrt{\frac{P_{return}}{P_{forward}}}=\frac{U_{return}}{U_{forward}}$$
The absolute value of the reflection coefficient, $|\Gamma |$, is the square-root of the ratio of return-power to forward-power, both now measured at one and the same location along the feedline (e.g., coax) to determine $SWR$. Equally, one can measure the forward voltage and return voltage at the same location, or their sum and their difference to determine $SWR$.
Yes, the $SWR$ can change along the transmission line. This is the case if the line is lossy. The $SWR$ is constant along the transmission line if the line is, or may be considered, lossless.
Assume we are at the end of the transmission line. The mismatch between load and characteristic impedance of the line determines the reflection coefficient and $|\Gamma |$, which is the square-root of the ratio of return-power to forward-power at the end of the feedline.
Now, we move away from the end of the line und closer to the feedline input; we consider a lossy line. At the new location, the forward-power is higher than the forward-power reaching the end of the line, due to the attenuation of the line. But the return-power at the new location is lower than measured at the end of the line because it has been attenuated along its path from the end of the line to the new location. Consequently, with lower return-power and higher forward-power, the ratio, $|\Gamma |$, i.e., the square-root of the ratio of return-power to forward-power at the new location is smaller than measured at end of the line. The smaller the $|\Gamma |$, the smaller and closer the $SWR$ gets to 1. (The same discussion can be presented in terms of forward and return voltages, of course.)
Your observation of $SWR$ changing along a (lossy) line confirms that $SWR$ measurements are best performed at the load if one wishes to determine the actual load mismatch.
For a lossy line, in a Smith-Chart the transformed load impedance moves along a spiral, which is the curve of the complex reflection coefficient, $\Gamma $. Asymptotically, the spiral merges into the center of the chart, as the distance to the load becomes longer and longer. At the center of the Smith-Chart, the transformed load impedance equals the characteristic impedance of the feedline; the reflection coefficient becomes zero, i.e., $\Gamma =0$ and consequently $SWR=1$. If the lossy line gets very long, at the input we will no longer feel the actual mismatch that might exist at the far end of the line. Therefore, the $SWR$ measurement at the input of a long feedline can significantly underestimate the $SWR$ given at the load side. Perhaps, in practice one might like to keep this in mind.
The following Smith chart shows an example of a very lossy line to see the effect. Example: 50Ω lossy feedline terminated into 200Ω (DP 1), i.e., $(V)SWR=4$ at load. Two circles of constant $(V)SWR$ (4 and 2) are shown in green.
In the above equations, all values $U_{forward}$, $U_{return}$, their sum $U_{forward}+U_{return}$, their difference $U_{forward}-U_{return}$, and the power values
of $P_{forward}$ and $P_{return}$ refer to the one location of the SWR-measurement. We can perform the measurement anywhere along the line (keeping in mind the measured $SWR$ on a lossy line decreases as we move away from load). We do also not need to relocate the meter for two separated measurements, provided we obtain by the applied meter-principle two values which allow to separate and to obtain, directly or indirectly, the forward component and the return component that both apply at the single location of our measurement at any spot selected along the line. With the knowledge of forward component and return component, we know the reflection coefficient and $|\Gamma |$ and, thus, we know $SWR$.
Now, assume the measurement principle allows us to obtain only the phase-dependent superposition of forward voltage and return voltage; this location-dependent superposition is the (transversal) voltage of the standing wave. From this single value, we are no longer able to separate the result into forward component and return component. Also, we might have no idea where our measurement location is relative to the phase of the standing wave on the line. However, in this case we can search for the positions of the maximum and minimum standing-wave voltages along the line, with the ratio of the measured voltages $(U_{max}/U_{min}$) being the SWR. (Any transmission line loss between the two measurement positions (separated by $\lambda /4$) is usually neglected.) But this measurement principle appears meanwhile as heritage from the past (photo; credit: QST-Magazine), although the principle might appear closer to the (original) definition of $SWR$.
