This is par for the course. I have friends who have had to manage manufacturing processes in China. You have to be very specific, to the point of utter paranoia, in the instructions that you provide. Wherever, and I mean absolutely wherever a corner can be cut, it will be cut at this level.
One friend had to get rubber household cleaning gloves made in China. He worked for a retailer and this would be for their store brand. If they specified something to be done, it would be done exactly the way they wanted. If something was unspecified, you could bet that the cheapest, shoddiest, most dangerous materials/processes would be used. For example, he never specified that the rubber used should actually be safe to be worn by humans. Prototypes came back with obscene levels of toxins in a glove that is meant to protect people from getting their hands dirty with household cleaning chemicals. There was just no common sense to fill in the blanks in the instructions here. Once they sorted out this issue, as another commenter mentioned, they would have to check each and every shipment because the manufacturer would try to sneak cheaper variants into the batches. Eventually my friend went and inspected the factory on a business trip only to find so many ethically questionable practices that he eventually decided to switch to another job.
If you choose to go the overseas route, I'd suggest that you find a consultant who has experience here. It will be well worth it. They'll be familiar with all the games that are played and will save you money.
As a westerner enjoying the luxuries of a western lifestyle, obviously I'd urge you to try to source/assemble locally. Cheap labour practices aside (i.e. I understand the argument that folks on that side of the world need work too and are willing to put up with cheap wages and long hours), a lot of the savings come from a complete disregard for the local environment. So these workers might benefit from the wages they earn, but they'll pay for it with their health. You probably wouldn't want heavy metals leeching into your community's drinking water, and you'd cease doing business with a factory doing so in your neck of the woods. But somehow folks seem to think it's acceptable if it's not in their backyard. That's an ethical question you'll have to answer for yourself.
All this said, quite frankly some of the manufacturing expertise simply no longer exists in North America. So your only choice is to get stuff done in China. I know a wealthy flag-waving patriotic ex-military businessman who confided in me as much about why he had to outsource overseas. He said it wasn't so much the cost but that the manufacturers overseas were simply better skilled. We like to think we've got superior technology/processes over here and that the only advantage of overseas manufacturing is the cheap price. But the more that manufacturing is done overseas, the less this will be true.