Mixing Winsor & Newton Pigment Markers & Copic Markers
About damaging the Markers.


I don’t see how you could, at most, just clean the tip afterwards if you use both types toghether. Just keep in mind, according to Amazon


  a smooth surface is important so as to no damage the nibs.



About using both types together.


The Winsor &amp; Newton Pigment Marker page only states


  Can I use Pigment Marker with other media, like water colour or oil?
  This is not recommended. The binder used in the formula of any given paint will block the Pigment Marker nib. However, Pigment Marker works with graphite pencils, and further experimentation is always encouraged. 
  Will Winsor &amp; Newton Pigment Marker™ work on standard bleedproof marker paper?
  Bleedproof paper is designed for dye-alcohol markers, which are extremely fast-drying and bleed through paper. Pigment Marker does not bleed through and is slower-drying; therefore, performance will be compromised due the paper’s high absorbency.


Since you should not use the same paper than the alcohol based markers the point is inferred.

The Blick seller of the W&amp;N markers  has definitive specs:


  Usage Note — Do not use with alcohol-based markers


However i found a Customer Review on Amazon


  Fantastic. I use this marker with alcohol based markers like copic/prismacolor as a way to add strong highlights. The texture of this marker is basically a paint consistency, so I have to be careful when using it this way. It works well for what I need, blends beautifully. Would recommend.


So, maybe you can try dual using the markers in non-essential works. 

However, my advice would be not to do it. Even within the same medium it is often problematic to use different brands because of the myriad of proprietary chemicals that could potentially interact.