An ILD is built for one fixed response (say, reaction at a support or moment at a section) by moving a unit load to different positions and recording the response each time.
So in an ILD, points remain fixed, position of load changes.
A third way to settle this is to look at what quantities are actually plotted on the two axes of an influence line diagram.
On the horizontal axis of an ILD, we plot the position of the unit load as it moves along the span. On the vertical axis, we plot the resulting value of one specific response (such as a particular reaction, shear, or moment) at one specific, predetermined location on the structure. Because the vertical axis is tied to a single, unchanging location by definition, that location cannot itself be a variable in the plot.
The very structure of the two axes used in an ILD confirms that only the load's position varies while the response point stays fixed.
Therefore, the correct answer is points remain fixed, position of load changes.