I've setup this scenario on Stellarium where the moon is aligned with East direction. I wanted to see the change of altitude of the moon if I move straight into its direction (time of observation is always the same) but instead it keeps changing its direction to the left. I expected to keep on the same azimuth and only change altitude but no, and I don't understand the geometric reason of it, for example If I walk in a straight line towards a building it won't move right or left. On the equator works as I expected.
I can approximate the scenario in question at 2020-03-15 04:23 UT.
At location 1, the Moon is at alt=34.6°, az=90.0°.
At location 2, the Moon is at alt=41.2°, az=85.1°.
We see a shift in azimuth because the Earth's surface is not cylindrical. You can get from location 1 to 2 by moving 721 km on a constant heading of 90°. However, the 36°S parallel curves almost 5° to the right in that distance.
Moving the same distance constantly toward the moon, along a great circle, would put you about 29 km north of location 2 with final heading 85.3°. Continuing around the world on the same great circle, you'd cross the equator at 39.6°E, heading 54.1°.
A view of the zenith showing the equatorial grid (blue) may help to illustrate the issue. The left half of the prime vertical (green) is due east. The longitude difference between locations 1 and 2 is equivalent to 0.53 hour of right ascension.