Alexander, this is very common, and is in fact normal behaviour for the type of SWR meter you are using.
For these SWR meters, the relative forward and reflected power readings change if the transmission line impedance changes. Even though the static characteristic impedance of the transmission line is fixed at say 50 ohms, as soon as you start transmitting, if the SWR isn't exactly 1:1, then standing waves appear on the transmission line, and this has the effect of changing the line impedance, or the dynamic impedance seen looking into the transmission line during operation.
If there are standing waves on the line due to the SWR not being perfect, then the line impedance becomes a function of the frequency you are using and a function of the length of transmission line. As you change frequency, the line impedance changes, and so the relative power readings on your SWR meter change. If you move the SWR meter along the transmission line whilst leaving the frequency constant (you normally wouldn't do this), then you get the same effect, and the line impedance and meter readings will change.
The SWR meter power readings are only accurate when the SWR is exactly 1:1. In this special case, the line impedance of the transmission line is equal to its characteristic impedance, because the characteristic impedance of the transmission line is equal to the impedance of the radio and of the antenna, and so there are no standing waves on the transmission line.
Having said all that, the SWR reading shown on the meter will always be accurate, because the SWR reading is not affected by the line impedance. The SWR reading is the ratio of forward to reflected power, which is directly related to the ratio of maximum and minimum amplitudes of the standing wave voltage, and ignoring losses, these ratios are determined solely by the impedance of the radio, transmission line characteristic impedance, and antenna impedance.
Hope that all makes sense!