The answer if @ProfM is already very complete, but I wanted to tackle your question from a more practical point of view.

The presence of imaginary frequencies indicate that there are atomic positions which are more energetically favorable at the ground state. So, the concept of "following" a mode means condensing it onto the reference structure, until you find the equilibrium positions.

To give an example, we can start with $BaTiO_3$ with $Pm\overline{3}m$ symmetry. By looking at the phonon frequencies you will notice the instability (imaginary frequency) at $\Gamma$. Which for this case in particular, corresponds to the so-called ferroelectric mode.

Once knowing the phonon of interest you can read its eigendisplacements ($U_{FE}$), and condense them onto the reference structure ($S_{ref}$) using different amplitudes $\alpha$, $$S_{\alpha} = S_{ref} + \alpha U_{FE}$$ And calculate the energy for each of the resulting structures ($S_{\alpha}$). You will notice that, as you increase the value $|\alpha|$, the total energy of the system decreases until it starts increasing again, creating a potential well. The structure with the minimum energy will be the one at equilibrium (according to the selected mode).

Coming back to the $BaTiO_3$ example, the new structure $S_{\alpha, E_{min}}$ should show a $P4mm$ symmetry. However, you would need to relax the phase, as strain also needs to be included.

Note: $\alpha$ can be any real value but it is case-specific (depending on the anharmonicities of the system and on the definition of the eigendisplacements), so would have to try with different options to determine which one is better for your case.

The answer of @ProfM is already very complete, but I wanted to tackle your question from a more practical point of view.

The presence of imaginary frequencies indicate that there are atomic positions which are more energetically favorable at the ground state. So, the concept of "following" a mode means condensing it onto the reference structure, until you find the equilibrium positions.

To give an example, we can start with $\ce{BaTiO_3}$ with $Pm\overline{3}m$ symmetry. By looking at the phonon frequencies you will notice the instability (imaginary frequency) at $\Gamma$, which for this case in particular corresponds to the so-called ferroelectric mode.

Once you know the phonon of interest, you can read its eigendisplacements ($U_{FE}$), condense them onto the reference structure ($S_{ref}$) using different amplitudes $\alpha$, $$S_{\alpha} = S_{ref} + \alpha U_{FE}$$ and then calculate the energy for each of the resulting structures ($S_{\alpha}$). You will notice that, as you increase the value $|\alpha|$, the total energy of the system decreases until it starts increasing again, creating a potential well. The structure with the minimum energy will be the one at equilibrium (according to the selected mode).

Coming back to the $\ce{BaTiO_3}$ example, the new structure $S_{\alpha, E_{min}}$ should show a $P4mm$ symmetry. However, you would need to relax the phase, as strain also needs to be included.

Note: $\alpha$ can be any real value but it is case-specific (depending on the anharmonicities of the system and on the definition of the eigendisplacements), so would have to try with different options to determine which one is better for your case.