Garmin Base Camp with Europe maps

No it's not. BaseCamp will normally plot routes across countries. You say you have a sd card for all of Europe. Are you using the map from there? If so where did the sd card come from and which map or maps are installed. Note BaseCamp can only access one map at a time, so it's possible that the sd card and its map are the issue 

The SD card is a genuine Garmin card purchased from the GPS store. It is installed in the card slot of my Nuvi 2539 lmt. The Nuvi is plugged into m Windows PC where Basecamp is installed. In "maps" I click on the City Navigator N/t Europe map. If I plot a route from Germany to France, I get a straight line. If I plot a route within Germany or France, for example, I get an actual route.  If I am viewing Germany, I see all the details but outside of the German borders I get no detail unless I move the map and re-click the Europe NT map. So, either I am doing something wrong or there is a problem with Basecamp. Any ideas?

I have not moved any of my maps. They are resident in the SD card.

No, it's not a problem with BaseCamp. If it's an original Garmin card with the full EU map installed then the map will be in 2 parts, EU South and EU North. This is because the maps are now so large the EU map has to be split in 2 or it would exceed the FAT32 limit. In BaseCamp you can either select North or South, so you will get a straight line if you try to plot a route across the North/South boundary. Those with the EU map installed on their device can install the full map to their PC using Express so don't have that issue with BaseCamp. Afraid the issue is the fact you have the maps on a SD card.

If Basecamp can only access one map at a time, how can I plot a route between two countries? Is the SD card a collections of different maps and only one can be accessed at a time?

I imported a KML file from Google maps that plots a route from Germany to France. Imported as a track into Basecamp it looks fine. When I try to convert to a route it turns into a straight line.

As I've already explained if Express had installed the map to your PC it's not an issue. You've bought a sd card where the EU map is split in 2. This isn't a problem when used on your device as it can read both maps at the same time. BaseCamp cannot do that, which is why you cannot create a route between the CN North map and the CN South map. By buying the sd card you are stuck with the issue.

In the end, it doesn't matter what the route looks like in BaseCamp; what is important is what it looks like on the device.

It is a bit inconvenient, but here a couple things you can try,

Using the map of Germany, build a route from your starting point to where you intend to cross the border, stopping on the Germany side. Change the profile of route to Direct.

Switch to the map of France and build a route starting on the France side of the border to your destination.  Change the profile to Direct.

Select the two routes and join them (list order should be from start to finish).  Open this new route and change the border crossing points don't alert points.

Send this route to your device and see how it routes on the device.

If it remains as a direct route, there is probably a way to change the profile of the route on the device.  But first, I would go back to BaseCamp, switch to the Global Map and change the profile to whatever you want.  Then send that to your device and see how it behaves.

If you plan on keeping the route in BaseCamp, keep it as a Direct route.  This will prevent any complications should you cause a recalculation of all your saved routes.

That's a good workaround but it does depend on the OP deciding where to cross instead of BaseCamp picking the route. Personally I feel the better option would be to create the route on the device and avoid BaseCamp routing altogether. BaseCamp could be used to send start and finish points plus any intermediate points required which would make creating the route on the device easier.