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

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

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

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.