Cut them hard in the summer, then again when they start to grow again about 6 weeks later, again hard. Have stock in the field throughout, preferably all or including sheep. Repeat next year and probably every four or five years.
If you want to spray, cut them first then when the new shoots are big enough go over with a weed wiper. You'll need to keep stock off while you spray and for a period afterwards - and be aware that the chemical is extremely persisitent and will continue to kill broadleaved plants in the excreted or muckspread dung of the animals that eat the treated reshes, whether they eat it wilted in the field or ensiled as silage next winter. Not a problem unless your muck ends up on vegetable areas ...
And yes, nothing is a long term solution unless you drain it - but keeping sheep and/or native ponies on and topping the reshes whenever they get tough and unappealing to the stock will keep them manageable.
If you have sheep and lamb them, remember that the reshes are mother nature's lamb creeps : the lambs will get a lot of shelter from them. So don't remove them all in fields where you will lamb or run young lambs.
I can't tell you about pigs' ploughing being a solution, I have wondered about it myself but would have to find a way to give the pigs some drier ground as well so they are not forced to be only on the wet, reshy ground. In any case, pretty much all our reshy ground is covered by one environmental scheme or another so we are restricted in what we are allowed to do - and ploughing, whether by pig or by machine, would not be acceptable!