Hi, I followed Twinstream's suggestions, but unfortunately without any effect. In addition I noticed that on the index page (with the RCS) an alternative icon is displayed, but on the other pages (without the RCS) a missing icon symbol appears. So, there must be something else here.
More searching tomorrow.

RCS doesn't have to be on every page to cause havoc with the glyph icons. It's enough that it is being used just once on the whole site.
Try one thing:
Make a copy of your source file, and remove everything pertaining to RCS. All the references, all the files. If the remaining source file is ok, then you will know which is the culprit. Then you probably need to bite the bullet and use the Carousel component.
And if you upload in between, you need to remove the RCS files from the server too.

Another thing I notice is that you are missing the fonts for the coffeecup icons to work. I do not see these in the font folder....

coffeecup-font-icons.eot
coffeecup-font-icons.svg
coffeecup-font-icons.ttf
coffeecup-font-icons.woff

Did you happen to delete them by mistake ? The icons are linked to the fonts.

I just got this reply from Suzanne:

'RCS was built with an old edition of icon moon icon pack. Newer SD uses the latest. Try removing the font resources for the embedded slider and see if it still happens.'

Is that what you have tried?

I think also the fonts for foundation icons will have to be added back in to the font folder as when custom fonts were added the foundation fonts may have been removed ?

Since the fonts folder for both SD4 icons and RCS icons have the exact names...

coffeecup-font-icons.eot
coffeecup-font-icons.svg
coffeecup-font-icons.ttf
coffeecup-font-icons.woff

whichever loads second will become the active icon set.

The only solution around this if one has a slideshow with icons, excluding slideshow controls which are svg but includes slide indicators which are icons, is to rename the fonts in the RCS font folder to something different like....

coffeecup-font-icons2.eot
coffeecup-font-icons2.svg
coffeecup-font-icons2.ttf
coffeecup-font-icons2.woff

Then in the RCS slideshow.css file
change all references to coffeecup-font-icons to coffeecup-font-icons2

That will allow the two programs to "coexist".

We helped another user not to long ago with this exact same thing.

Re-uploading the fonts (thank you Twinfield!) solved the issue for al pages except the index (the page with the RCS). Last night I have quickly tried to remove all RCS files and references, but that made no difference. I guess I will have to look deeper and it may be even better to replace the RCS with the carrousel component. But this will have to wait until after the weekend. Thank you very much all for your tips!

Hi, yesterday when I replied to you saying you were originally using Bootstrap3, I was also trying to help someone who actually did use BS3, so I mixed the two of you up. Sorry about that.
You are of course using Foundation, and the slideshow component for Foundation is called Orbit. That is the one I'd recommend using.

It appears to me that your font folder for your slideshow is still up on your server. You will need to delete that folder on the server side or rename the font files to something different than "coffeecup-font-icons" so it does not conflict with foundations icon set which uses the same file names.

https://www.persoonlijkrome.nl/Home-slider/fonts/ <delete>