From polygon to continuous measure of diversity?

First of all; I have tried to find a similar question, without success. Maybe it's because I'm quite new to GIS and I don't really know what exactly I'm looking for. If someone points me to a similar problem, I'd be happy to remove this post.

I need to create a 'continuous' or raster (in small grid cells) variable of population diversity for a given country. I have a shapefile showing the spread of the ethnic groups in polygons (fig. 1), and the outcome I'm looking for is a the 'mean indicator of diversity' in each of the administrative units (AU's, in this case, the 360 Nigerian constituencies).

Fig 1. Population groups polygons in Nigeria

The solution I came up with was to get the area percentage of each polygon in each AU, and calculate a heterogeneity index from that. But the problem is that I would be leaving aside quite a lot of information due to the distribution of administrative units. As shown in the fig. 2, the squares 'a', 'b', and 'c' would have the same 'segregation index', but it is clear that they are not in the same position vis-a-vis the 'hot spots'.

Fig 2.

So I though that another solutions could be create a grid map, and calculate the distance to the nearest border, but again sharing only one border is not the same as being in the central part of the map, where several groups live together.

After finding this question, I guess that the polygons could be transformed into points using their centroids, and then apply the same method. But the truth is that I'm new to this, and that question is not really clearly answered. How could I made such a thing?

Using another example, I want to create something like this (images from this website):

Given the distribution of some points with different qualitative characteristics, get a measure of diversity from where I could estimate a the 'mean heterogeneity' of each administrative unit.

How could I do it? I use R and QGIS, so I don't mind in which platform is the solution based on.