I think your confusion is in the way you are thinking... a "source" rather it be a battery or charge board or power adapter or whatever does "put out" current. Current is drawn. The voltage it's what's "put out".
Think of it like your water faucet. The pressure is like voltage. If your supply is 30psi then that pressure is there, even if all you faucets are turned off.The flow of the water is current, actually current is a measure of the flow of electricity. It's only when you open one of your faucets and "request" a certain amount of current that it actually flows. Now if you have a gigantic faucet capable of delivering 50 gallons per minute, just because you open that faucet doesn't mean you're going to get that current. You can crack open the faucet just a little bit, and only a small current will flow.
You can't think of the charge board "forcing" a current, it's only the circuit that will "draw" whatever current it needs.
On the other hand, if you try to get 50 gallons per minute out of your faucet that can only deliver 1 gallon per minute, nothing bad happens, it just won't give you the amount of flow you want.
When the charger is hooked up with the battery and turned on, the battery because of it's internal resistance which is very low will try to equalize the voltage difference between it's current voltage (let's say 3.5 volts) and the charger voltage of 4.2 volts. But the charger circuit limits the current, so it can only get say .5A or 1A (typical of the charge board we're talking about here). So that's all the "flow" that will go into the battery.
Since the charge board is hooked up across the battery, and so is the load (in this case the dna chip) any current the dna tries to draw will be facilitated by the charge board. I won't go into the math, but let's just say that since the charge board is at 4.2v and the battery is at 3.5v a portion of the current supplied to the dna will come from the charge board. But if you fire the device, and it starts drawing say 8A, that's way above the .5 or 1A the charge board can supply, so it will mostly be the battery supplying that current.
So let's say I went all out and put a massive charge board that was capable of delivering 500A of current... If I fire the device and it draws 8A it still won't hurt anything, because it's only going to draw 8A...
Again this is because the load "draws" the current, the source doesn't "force" it.
One last thought to maybe try to help you understand...
If I'm thirsty and I get some water... It doesn't matter if I get a 12oz bottle of water, or find a big river. I'm only going to "drink" what I want.. Just because I drink from a river doesn't mean the entire river is going to get jammed into my throat.