As I get older, I'm more likely to need some kind of medical treatment while travelling. My current healthcare policy appears to cover such treatment, but I'm looking around for a less expensive alternative. Also, we are starting to dream about moving to Europe for a few years so we could travel more extensively.

What healthcare policies are best for the traveller, and the expatriate?

If you live permanently (= have a residence permit) in any EU country, that country's healthcare covers also you. So you don't need some special healthcare policy, you are already covered.

For traveller, just take a normal travel insurance (one that covers also possible hospitalization).

A US citizen without EU passport would have to provide their own health insurance before being granted residence permit. And even with a EU passport, if you haven't contributed to health insurance funds in the country (or any other country with reciprocal agreement, but that excludes US), you may still not be eligible.
There are rather expensive ex-pat policies, but they are aimed at international executives and the cost is almost always met by their employer. An individual policy covering an older person with existing medical conditions would be seriously costly. In Spain, for example, EU pensioners get free health cover. And UK has a National Health Service that gives free treatment (with certain exceptions) to all residents, regardless of nationality.

Well, it's sort of free. People pay through taxes.

CB: I don't get it,you say your current policy covers you,then you ask for a less expensive alternative, so what are you talking about.

Just a reminder, the American Hospital in Paris accepts BC/BS. Keep that in mind if your travels take you to Paris and you need medical care.

My current policy is very good for families with children, but our kids are grown now, and the premiums keep going up, so I am looking for a more economical alternative, just the way I am looking for a lower airfare for our next trip.

I doubt this forum is the best place to find a cheaper healthcare policy.I also would not pick my policy depending on if it covers me on vacation in the off chance I may need it.When we travel I get a specific comprehensive policy for each trip ie:family of 4 two weeks total coverage approx$115.00 from my insurance agent.This incl. emergency medevac etc.etc.