In Special Relativity, you have to be very careful what you mean by something like "the relative tick rate".
Lets look at our two different scenarios from the point of view of discrete jumps, instead of continuous. That might help explain what's going on.
Instead of firing the rocket continuously, assume that the way the acceleration works is that there is a schedule: At time t=0, rockets are fired to accelerate to speed 10% the speed of light. At time t=1 (according to the clocks on board the spaceships), rockets are fired again to accelerate to 10% of the speed of light relative to the first speed. Etc.
So let's assume that the initial distance between the spaceships is ##L##.
The rear spaceship fires its rockets at event ##e_1## with coordinates ##(x_1, t_1)## (using the coordinates of the inertial frame of the larger ship).
The second spaceship fires its rockets at event ##e_2## with coordinates ##(x_2 = x_1+L, t_2 = t_1)##. (Same time, different location.)
Now, after accelerating, the spaceships are (momentarily, until the rockets fire again) at rest in a new frame. This new frame has a different coordinate system, ##x', t'## related to the first coordinate system through:
##x' = \gamma (x - vt)##
##t' = \gamma (t - \frac{vx}{c^2})##
where ##v## is 10% of the speed of light. So in this new coordinate system, ##e_1## has the coordinates:
##x_1' = \gamma (x_1 - v t_1)##
##t_1' = \gamma (t_1 - \frac{v x_1}{c^2})##
##e_2## has the coordinates:
##x_2' = \gamma (x_2 - v t_2) = \gamma (x_1 + L - v t_1)##
##t_2' = \gamma (t_2 - \frac{v x_2}{c^2}) = \gamma (t_1 - \frac{v x_1}{c^2} - \frac{v L}{c^2})##
Now, if we subtract the coordinates, we get:
##\Delta x' = x_2' - x_1' = \gamma L##
##\Delta t' = t_2' - t_1' = - \gamma \frac{vL}{c^2}##
Note: In this new reference frame, we find two weird things:
- The distance between the rockets has grown from ##L## to ##\gamma L##.
- The two rocket firings were not simultaneous. Since ##\Delta t' < 0##, that means that, according to this new reference frame, the front rocket fired earlier than the rear rocket. What that means is that the way things look in this new frame, first the front rocket fires, when its clock shows time ##t_1##. Then a time ##\Delta t'## later, the rear rocket fires when its clock shows time ##t_1##. So in this frame, the clock in the front rocket is ahead of the clock in the rear rocket by an amount ##\Delta t'##, since that's how long it has been at rest in this frame waiting for the rear rocket to fire.
So at this point, you can see that pattern: If every second according to the clock aboard the two rockets, the rockets fire, then the rockets will drift farther and farther apart (as measured by those aboard the rockets) and the clock in the front rocket will get farther and farther ahead, also.
If the two rockets want to keep the same distance, then it's necessary for the front rocket to fire either less frequently, or with less intensity.