You think big, I'll grant you that.
This is a partial answer, probably should be a comment, but it's too damn big.
Factor 1: The ocean is large. 1.3 billion cubic kilometers. Most of the disolved minerals are present in very small numbers. You are going to have to process cubic km of water. I suspect that the only way to do this economically is to develop materials that bind to a particular atom or ion, coat fabric with it, and let the currents go past it. For perspective the larger open pit mines have total volumes measured in a small number of cubic kilometers.
Factor 2: To gather material at a level that would make a measurable decrease in the mineral content, you would have to process a measurable fraction of the ocean. Suppose that you decided to mine the Gulf stream. So you intercept a 300 km wide strip of ocean 100 m deep. The gulf stream moves at about 4 mph -- 6 kph say. So your system will intercept 180 cubic kilometers of water per hour. At that rate it would take you 7 million hours (about 8 centuries) to process the entire ocean once.
Factor 3: Replenishment processes. Most minerals are not close to saturation. Common salt, the most common mineral, is present at about 3%. Saturation for salt is about 30%.
There will be some exchange between sea floor sediment, but this is going to be VERY slow. Turnover times for the abyssal plain are on the order of 10,000 years.
I suspect that the primary replenishment processes will be geological -- erosion. If this is correct, then you would do far better by filtering the rivers as they flow to the sea, mining our cities sewage (meta content thousands of times higher than in sea water) ANY technology that can extract useful minerals out of sea water can extract them out of any liquid waste stream, with likely far greater efficiency.
Intercepting significant amounts of an ocean current will have climate effects. People in nothern Europe will NOT be pleased with you if you slow down the Gulf stream by 10% Slowing it down, also means that the temperature differential between the tropics and the pole will increase giving greater potential energy to hurricanes. You just picked up another hate group.
Caveats: Even now vacuuming up manganese nodules off the ocean floor is worth while. There are likely conditons that have concentrated other minerals in sea floor sediments. But I didn't understand this to be the type of mining you meant.