dimensional analysis - How to introduce dimensionality in a dimensionless framework?

This question is an extension of this one. I have been told that to introduce dimensionality in a dimensionless quantity I need to multiply with suitable parameters. For instance, for velocity I have to: $$v'=v*(l/\tau)$$ where $v$ is the dimensionless velocity and $l$ is the step length and $\tau$ is the time step. But the reference I am using Random walks of molecular motors arising from diffusional encounters with immobilized filaments defines $$v=1-\gamma-\delta-0.5\epsilon$$ and the units of $\epsilon'$ is $\tau^{-1}$. where $$\epsilon'=\epsilon*\tau^{-1}$$.

My question is how all of this makes sense in dimensionality. In the exact dimensional analysis, we are adding quantities with dimensions $\tau^{-1}$ and getting a dimensional quantity of $l/\tau$. Furthermore, the diffusion coefficient in the same reference has been defined as: $$D=v^2/\epsilon^2$$ Now if I want dimensionality of $D'$ I will have to do: $$D'=D*l^2/\tau$$ However, If I use the dimensional quantities $v'$ and $\epsilon'$, the dimensionality for $$D'_\text{wrong}=v'^2/\epsilon'^2$$ will be $\frac{l^2}{\tau^2}*{\tau^2}=l^2$, which is wrong. Also to get proper dimensionality the last analysis suggests that $\epsilon'=\epsilon \tau^{-0.5}$ which is different than the aforementioned analysis. I am confused about why I am getting these inconsistencies.