transporting precious metals to EC

You can ship them, but its very risky.  The box(es) would be opened several times, and who knows just how sticky some of the fingers are.  The rules around that movement are from the U.S. and are available through US Customs, either their website or by letter.  Basically, as long as the face value of the coins is less than $10K, you're OK.
Personally, I'd carry my metals, even if it meant a separate trip.

Separate trip is right.  Keep your metals in a safe U.S. institution for now.

Until you're established in South America .. why expose yourself to the risks of theft, logistical hassles, and taxes on your asset if you exit Ecuador or wherever...

If Ecuador doesn't work for your lifestyle or your health, and you move on after a year, you could face the same three problems again.

cccmedia in Quito

Thanks  Harry.   
So, maybe the banks are no safer than  the ones in the states.   things are changing.
Ecuador Visa's lawyer  said,  just "declare".   Appreciate your advice !

Thank you very much for your quick and wise answer.   
Bless you , ccc!