Transferring flows (connections + values) between polygons

You could do a join between the three layers, then aggregate by layerB. Virtual layers can likely be used. I am not sure if the important data is in layerA or in the flow layer.. Here is an (untested) possibility:

SELECT b.id, b.geometry, sum(a.myVar)
FROM layerB b
LEFT JOIN flow f
   ON ST_Intersects(ST_EndPoint(f.geometry),b.geometry)
 JOIN layerA a
   ON ST_Intersects(ST_StartPoint(f.geometry),a.geometry)
GROUP BY b.id

With the Virtual Layers, theoretically, it's possible (with shapefiles, the process will be extra long, but if the layers are in a Spatial Database, I think it is a lot faster).

Here the code :

WITH inter_ab AS ( 
--create intersection between LayerA and LayerB 
SELECT LayerA.id || '_' || LayerB.FLAECHEID AS id, 
LayerA.id AS id_a, 
ST_AREA(LayerA.geometry) AS area_a, 
LayerB.FLAECHEID AS id_b, 
ST_INTERSECTION(LayerB.geometry, LayerA.geometry) AS geom 
FROM LayerA, LayerB 
WHERE ST_INTERSECTION(layerB.geometry, layerA.geometry) IS NOT NULL 
),

--calculation of the new flux value 
new_flux AS (SELECT t1.id_b AS origine, 
t2.id_b AS dest, 
SUM(Flows.flux * ST_AREA(t1.geom) / t1.area_a * ST_AREA(t2.geom) / t2.area_a) AS value  
FROM inter_ab t1, inter_ab t2, flows 
-- no connection between the same feature 
WHERE t1.id <> t2.id 
-- rule 1 
AND t1.id_a <> t2.id_a 
-- rule 2 
AND t1.id_b <> t2.id_b 
-- get flow data 
AND flows.origine = t1.id_a 
AND flows.dest = t2.id_a 
GROUP BY t1.id_b, t2.id_b
)

--create flows between original layerB features
SELECT new_flux.origine,
new_flux.dest,
new_flux.value AS flux,
make_line(ST_CENTROID(t3.geometry), ST_CENTROID(t4.geometry)) AS geom --ST_MakeLine under postGIS
FROM LayerB t3,
LayerB t4,
new_flux
WHERE t3.FLAECHEID = new_flux.origine
AND t4.FLAECHEID = new_flux.dest

The graphical output will look like

Output

The result was tested manually. The difference in "FLUX" values is neglectable.

The final output will inherit styles from "Flow" and look like

Output_Final

I recommend to test it with a few data, and if it takes too long for large data sets, execute step by step the queries ("inter_ab", "new_flux") and save the result and execute the next query.