This isn't a question with one clear answer, and this problem is one of the big reasons why larger companies trying to outsource to China will always send somebody to oversee the process locally.
In general, this is not a question of 'how to handle these matters', but it is a question of specifically which regulations you need to comply with. Before you start your first zero series, you let your lawyer figure out exactly which regulations there are for your electronic product in the target market countries. There are specialized lawyers for this kind of stuff and they are worth their weight in gold. Things can get very complicated when you want to sell a product over multiple continents, so you're probably going to spin multiple variants of your product to account for that.
Once you have a clear list of regulations to observe, you first make sure you find an assembly house that is certified according to ISO9001:14001. This is a very formal way of saying that such a company won't risk sourcing shoddy shenzhen market components to shave off a few cents, and that the company will act in accordance with your formal wishes. Then you give them a distilled-down version of all the specific things you wish them to do for you to comply with all the regulations. For instance, for the Netherlands:
- All components are RoHS, halogen free. Any replacement or revision is communicated and tested before substitution in the assembly line.
- Flux and solder shall be used in accordance with RoHS
- Component alignment is in accordance with [forgot the ISO-number], and you usually supply a document with inspection guidelines
- Boards shall be washed at least twice until no visible residue is left, with a washing agent that doesn't leave toxic substances, either from the washing agent or flux residue
- Board material is UL 94-V rated, UL number and flammability rating are printed in copper and silkscreen in a location visible to an external inspection agency (e.g. TNO/KEMA here, FCC in the US)
- Packaging is designed in such a way to safely transport lithium ion batteries separate and non-functional (e.g. with sticky plastic covering), charged at most 80%, air shipment is insured and liability is fully covered
Just an example, this is close to all you need to tell the assembler and packager. Last time I was involved with production it was literally just 2 or 3 pages of instructions plus the assurance that we were dealing with somebody who would actually read and understand this (which you will verify with the zero series production). ISO9001:14001 helps a lot, if a company doesn't comply they can be punished by the auditors and they don't want to risk that (it's a pretty expensive audit).
Most other regulations are dealt with on other levels: component choice, layout, electromechanical product design, packaging, shipping, sometimes obsolescence planning and warranty. You won't need to communicate that with your production partner. The examples you give on export restrictions are something you deal with as a design engineer more than anything else; well-known examples are FPGAs, PV solar cells, microbolometers, photon multiplier tubes and radioactive substances.