Consider a 1-D system of thickness $L$ (a slab) with conduction coefficient $k$, density $\rho$, and specific heat capacity $\tilde{C}_p$. It is subjected to a sudden temperature change from $T_o$ to $T_\infty$ on one side (the left side). The convection coefficient on that side is $h$. The resulting formulation and boundary conditions are developed in this example PowerPoint posting.
$$ \rho \tilde{C}_p\frac{\partial T}{\partial t} = k \frac{\partial^2 T}{\partial x^2}$$
We can make this dimensionless and allow an unrealized temperature to be defined as $\theta = (T(z,t) - T_o)/(T_\infty - T_o)$ where $T_o$ is the temperature at time $t = 0$ throughout the entire slab. In dimensionless form, the temperature profile through the slab as a function of time and position is determined by the equivalent of Fourier's second law. A defining time constant of the system is found from the Fourier number
$$ N_{Fo} = \frac{k t}{\rho \tilde{C}_p L^2} $$
The Fourier number can be viewed essentially as the dimensionless time needed to reach a specific temperature $T$ at a specific position $x$ in the slab.
You have two slabs A and B with $L_B > L_A$. For all else the same, $N_{Fo,A} < N_{Fo,B}$. This means, at the same relative position $Z = z/L$, slab A will reach temperature $T$ sooner than slab B. Alternatively, at the same time $t$ for the same distance $x$ from the surface exposed to $T_\infty$, slab A will be at a temperature $T_A$ closer to $T_\infty$ while slab B will be at a temperature $T_B$ closer to $T_o$. The opposite is true at the position closer to the side of the slab that is held at $T_o$.
The exact solution requires separation of variables and is shown in the PowerPoint link. A simpler analysis is to show the case for unrealized temperature as a function of dimensionless position in a slab with the left wall held at $T_\infty$ and the right wall at $T_o$. Such a sketch is shown below.
At the instant $T_\infty$ is applied on the left wall, the profile is abrupt at that wall and $T = T_o$ throughout. The left is at an unrealized temperature $\theta = 1$ and the right is at $\theta = 0$. As time progresses, the temperature profile develops roughly as shown. At steady state (infinite time), the profile is a straight line between $\theta = 1$ on the left and $\theta = 0$ on the right.
Consider that we are half way in slab A. This is the solid vertical line in the picture. Now consider slab B with $L_B = 2L_A$. The same absolute distance into slab B from the left side is marked as B1 in the picture. We see for the same time, the unrealized temperature in slab B is always higher at that absolute position than at the same absolute distance from the left wall in slab A. Now consider a position that is the same relative distance from the right side in slab B. This is B2. We see for the same time, the unrealized temperature in slab B is always lower at that absolute position than at the same absolute distance from the right wall in slab A.
