Here's a spacetime diagram drawn on rotated graph paper.
(All light-clock-diamonds have the same area
and all light-clock-diamonds have lightlike edges, by Lorentz invariance.)

Note that, in addition to time-dilation,
there is length-contraction and relativity-of-simultaneity
(which are not as easily represented on your spatial boxcar drawings).

I chose nice numbers to make the arithmetic simple.
Let Bob travel at (3/5)c to the right on Alice's diagram,
with boxcars spaced (for convenience) 3 of Bob's sticks apart.
Alice is at rest, with boxcars spaced identically with 3 of Alice's sticks.

When the trains first meet, the lead clocks read 0.
In Alice's frame, when her clock reads 0, all of Alice's boxcar-clocks also read 0.
In Bob's frame, when his clock reads 0, all of Bob's boxcar-clocks also read 0.

I think you should be able to read off all of the clock readings when the various boxcars meet.

Notice a bunch of Minkowski-right-triangles similar to
a 3-5-4 Minkowski-right-triangle [which is characteristic of relative-velocity (3/5)c].

Notice that, for v=(3/5)c, the Doppler factor is k=2.
The Doppler factor is an eigenvalue of the Lorentz boost.
So, to get Bob's diamond from Alice's diamond,
stretch by k=2 along the forward future-lightlike direction, and
shrink by k=2 (to preserve area) along the backward future-lightlike direction.

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And here's the (4/5)c case, with boxcar spacing 4 sticks,
4-5-3 Minkowski-right-triangles, and Doppler k=3.

These spacetime-diagrams on rotated graph paper are essentially
"spacetime-diagrams decorated by the light-signals in a ticking light clock",
which provide a construction for doing graphical calculations in special relativity.

Essentially, once you know what the light-clock-diamonds are,
use them like coordinate-bricks to lay out
so many ticks-of-time and so many sticks-of-space.

If you choose nice-numbers (velocities with rational Doppler-k factors),
and convenient choices based on the associated pythagorean triple,
you can lay out diagrams to count off your answers.
Once you've developed your geometric and physical understanding and intuition,
you can better appreciate the associated formulas (often involving right-triangle leg-ratios as hyperbolic-trigonometric functions) as applied to whatever values you are given in the problem.

Upon looking for certain triangles,
these diagrams will summarize and support any explanation you may wish to use:
time-dilation, length-contraction, doppler, lorentz-transformation,
some combination, etc.