The methods of propagation match those for reproduction outlined on the genus page.
If a plant has been identified as having offsets, ( see reproduction from rhizomes), the tuber should be dug up after it's gone dormant and the offset removed with a sharp knife. The cut surfaces should be treated with 'Flowers of Sulphur' or a similar powder fungicide.The offsets are then planted like any other tuber, (see Cultivation).
When a stoloniferous plant has formed a clump, it can be divided when it's gone dormant and the sections replanted. Don't be too greedy: I once destroyed a Campanula by cutting it up into too many small pieces!
Fertilized flowers develop into green berries which slowly turn red as they ripen.
As with other genera, when germininating seeds, I try to mimic what happens in nature. The seeds are sown in autumn when the plants go dormant. The red berries are squashed under running water in a stainless steel sieve with the back of a wooden spoon until all the flesh has been removed. They are then planted in a deep seed tray with a thin layer of compost on top and left until spring. It's important to wear latex gloves whilst doing this as the fruit contains a skin irritant. 
In their first year, some species will only grow a tuber without any leaves. I keep a tray of seeds for at least two years; potting them up when the tuber is large enought to handle.
At Redhall, species which flower early in the year can set fruit which will ripen by October and can then have the seed extracted as above for sowing. However, the later flowering species don't get past the green stage before the stem collapses as the plant goes dormant. We are experimenting to counter this by cutting off the stems before they collapse and keeping them in water in the conservatory. This should allow the berries to ripen but we've yet to see if the seed is fertile.
In 2018 we tried this with two sets of fruit from A. consanguineum; one which eventually ripened on October 25th, the second on November 11th. The seeds were cleaned and sown in deep trays at that time. The first set began to germinate in the middle of May, 2019; the second in the middle of June. The germination was somewhat limited, (~20 from a batch of about 50 seeds from the October set but ~40 from for the November sowing), but the method does work.