I first tried this route when my cousin was in town visiting.  I wanted to take her "hiking" to see the Swiss countryside, however wasn't comfortable using the soft baby carrier for such long periods of time and had not yet purchased a framed hiking backpack.  So, we found this route which is along the Swiss National Route #4 - Via Jacobi.  The national route crosses the entire country and is marked by the yellow signs with the number 4. 
This section of the trail conveniently uses the street my house is on as part of its path from Coppet.  I have now done this route twice, the second time with a friend who also pushed a stroller.  The trail can easily be broken into smaller segments by taking public transportation from any of the towns you walk through.  We walked from Coppet to Nyon and then took the train back to Coppet.  The walking portion of this hike took us about 3 hours not including some stops to allow the children to stretch their legs (such as in the stream we passed just before entering into Nyon).
I first found this route while doing a simple google search and found the route on My Switzerland's website.  I have since used Switzerland Mobility's website and have downloaded the app onto my phone.  With the app you can track yourself along their maps to see if you are still on the trail.  This is very helpful as I have accidentally gotten off the trail a few times and have used the app to help find my way again.
The walk passes through the charming villages of Coppet, Founex, Céligny, Crans près Céligny and ends in Nyon.  Between the towns the walk includes paths through forests, streets next to vineyards and apple orchards, views of churches, castles and neighborhoods with magnificent flowering and well manicured landscapes.
 The walk was large wheel stroller friendly the entire way with one set of steps in a foresty area that were managable with the stroller as they were deep and not too high.  The strollers could be "stepped" up and then down by tilting the front wheel up and then pushing until the back wheel was up and vice-versa.  Photos of the steps below:
 This was the only area with steps and they didn't require lifting/carrying the strollers, so I consider them stroller friendly.  There were a few shaded areas through the woods that felt nice in the heat.  During these times we let the children out of their strollers to do some of the walking themselves (also because some of the walk is along the side of streets or through towns so the forests were the safest areas for them to run around).
The walk ends in beautiful Nyon where it is easy to find a refreshment among its numerous restaurants.
To view more photos of this hike click here