It's taken me nearly 8 years but I finally found the answer!

1. Open Field Calculator to update field "VALUE_A" in Layer A, and use the following expression:

• aggregate('Layer B', 'sum', "VALUE_B", within(centroid($geometry), geometry(@parent)))
  

......................

Use Show Help to learn other functions you can use in each of the arguments:

• aggregate(layer,aggregate,expression[,filter][,concatenator=''][,order_by])
  

I have found this does time out for larger datasets, if this is an issue then you can select a subset of overlapping polygons in both Layer A and Layer B, check 'Only update selected features' (in Layer A), and use the following modified expression:

• aggregate('Layer B', 'sum', "VALUE_B", is_selected('Layer B', @feature) AND within(centroid($geometry), geometry(@parent)))
  

Finally, if you wanted to update with a text value from the containing polygon you can use 'min' as the aggregate instead of 'sum'.

:)

It's taken me nearly 8 years but I finally found the answer!

Open Field Calculator to update field "VALUE_A" in Layer A, and use the following expression:

aggregate('Layer B', 'sum', "VALUE_B", within(centroid($geometry), geometry(@parent)))

Use Show Help to learn other functions you can use in each of the arguments:

aggregate(layer,aggregate,expression[,filter][,concatenator=''][,order_by])

I have found this does time out for larger datasets, if this is an issue then you can select a subset of overlapping polygons in both Layer A and Layer B, check 'Only update selected features' (in Layer A), and use the following modified expression:

aggregate('Layer B', 'sum', "VALUE_B", is_selected('Layer B', @feature) AND within(centroid($geometry), geometry(@parent)))

Finally, if you wanted to update with a text value from the containing polygon you can use 'min' as the aggregate instead of 'sum'.