If an object at t=0 begins to free-fall into a black hole from X in Kruskal coordinates (https://en.wikipedia.org/wiki/Kruskal%E2%80%93Szekeres_coordinates), what does its path on the Kruskal-Szekeres diagram look like? Is it a hyperbola, or a straight line, or something else?
It looks like this picture on the right from Misner-Thorne-Wheeler:
It's almost a straight line.
But can I add this: I don't like Kruskal-Szekeres coordinates at all. Take a look at the picture on the left. That shows the object's path on the Schwarzschild diagram. Note how it's truncated vertically? The vertical axis is the time axis. In order to cross the event horizon at r=2M, the infalling body allegedly goes to the end of time and back$^*$. The horizontal axis is the spatial axis. Look across from right to left near where it says τ=33.3M. The infalling body is in two places at once, just like the elephant and the event horizon. Also note where the Wikipedia Kruskal-Szekeres article says this: "Note GM is the gravitational constant multiplied by the Schwarzschild mass parameter, and this article is using units where c = 1". That just doesn't square with what Einstein said: "As a simple geometric consideration shows, the curvature of light rays occurs only in spaces where the speed of light is spatially variable". You'll be aware that the coordinate speed of light at the event horizon is zero? So a light clock stops? IMHO Kruskal-Szekeres airbrush over this by effectively putting a stopped observer in front of a stopped light-clock and claiming he still sees it ticking normally.
$*$ Seriously. You can even read about this in the mathspages Formation and Growth of Black Holes: "the infalling object traverses through infinite coordinate time in order to reach the event horizon, and then traverses back through an infinite range of coordinate times until reaching r = 0 (in the interior) in a net coordinate time that is not too different from the elapsed proper time".