My wife is Ukrainian. We sailed around France, Netherlands, Germany and Denmark last summer. The issue is not customs clearance, but immigration clearance. We were keen to play "by the book" so as not to blot her copy book so far as the immigration auhorities are concerned.
Obviously, the non-national needs the relevant visas (a "Schengen" visa covers much of Europe, but strictly speaking you should apply to the embassy of the country you're going to visit first).
For leaving the UK, we contacted various authorities, including the police, Foreign Office etc to find out if she should get an exit stamp in her passport, and, if so, how (we were leaving from Lymington). We were advised that there is no need for this or procedure outside major airports etc, and we should just leave.
In arriving in a foreign port for the first time, we always flew the "Q flag" so that no one could say we were trying to sneak in ('though people probably thought I was daft flying "Q" on a British yacht). Generally, this attracted zero interest from Customs, even in Customs ports.
The exceptions were Netherlands and Germany. Arriving IJmuiden from Harwich we were visited by immigration first thing in the morning (even though I hadn't put up the yellow flag), and passports were inspected. My wife had the Schengen visa, s there was no problem.
Second on the River Eider in Germany we were hunted down by immigration in the middle of nowhere (we had mentioned in the bar the night before that my wife was Ukrainian) and had the boat thoroughly checked and our passports taken for verification. Again, having the relevant visa, they were returned some hours later with no problem.
Incidentally, we took a ferry to visit England from France. The French immigration officers were very interested as to how she entered the country without an entry stamp, but were quite satisfied when we told them we had sailed across to St. Vaast on our own boat.
It may be that strictly speaking you should enter the country concerned at a Customs/immigration clearance port, which would be very limiting, but in practice none of the authorities seemmed to regard that as important provided your papers are in order.
You must clear in with Customs at your first port of call in the EU. For many US boats this will be Horta, in the Azores. Beware of stopping over at off-lying islands like the Scilly Islands, where there are no customs facilities, without clearing in properly first, as a watch is kept for arriving US flagged yachts.
Once cleared in the EU, that's all you need. (If for any reason VAT is payable, that can't be done in the Azores incidentally; but that should not apply to you).
But beware! Some suprising bits of the European Community are not in the EU and there you will have to check in with Customs again, and then at your next port of call in the EU. The UK Channel Islands (Guernsey, Jersey), and the Canary Islands are examples.
You will be permitted to stay in the EU for a maximum of 6 months, but it should be easy to get that extended. Beware becoming liable for VAT (if Customs have reason to suspect you are importing your yacht).
Incidentally on immigration, some spots still seem to run to their own rules. For example, I think you still need to check in with police on each Azores island you visit. (The trick sometimes is to find them - try the local bar).