How do identical distributions "except for a shift in centrality" look like? In this question, someone mentions that a Kruskal Wallis test can be interpreted as a test of median differences, provided that...

"you can assume that rate is continuous and the distributions for all groups are identical except for a shift in centrality".

My knowledge in statistics is not very advanced and I am wondering what this means. I don't understand how a change in centrality (meaning a change in mean / median, right?) can not change the distribution type.
 A: This is an oversimplification, but still an example:
$\mathcal{N_1}(\mu_1,\sigma_1^2)$
Consider two normal distributions, defined by $\mathcal{N_1}(\mu_1,\sigma_1^2)$  and $\mathcal{N_2}(\mu_2,\sigma_2^2)$.
If you have $\mu_1=\mu_2$ and $\sigma_1=\sigma_2$ the two are identical.
However, if $\mu_1\neq\mu_2$, they are not identical anymore, because their centres are different. This is what one could call "identical except for a shift in centrality" in a more colloquial language.
A: 
The first case is one where both distributions are identical except for a shift. (Since they are otherwise identical, the additional "in centrality" is redundant.) In the second case, we have the same type of distribution (both are normal), but they are not only shifted, their variances also differ. In the third case, we have different types of distribution, a normal and a $\chi^2$.
R code:
xx <- seq(-5,5,by=.01)
par(las=1,mfrow=c(3,1),mai=c(.5,.5,.5,.1))

plot(xx,dnorm(xx,-1),type="l",lwd=2,xlab="",ylab="",main="Shift in centrality")
lines(xx,dnorm(xx,1),lwd=2,col="red")

plot(xx,dnorm(xx,-1),type="l",lwd=2,xlab="",ylab="",main="Shift in centrality and different variance")
lines(xx,dnorm(xx,1,2),lwd=2,col="red")

plot(xx,dnorm(xx,-1),type="l",lwd=2,xlab="",ylab="",main="Different distributions",ylim=c(0,0.6))
lines(xx,dchisq(xx,2),lwd=2,col="red")

