When wiring EC-3 connectors, do the wires go through the blue plastic and the ends are then soldered or are the wires soldered first and then pushed into the blue holders?

You have to solder the wires to the connectors first. Otherwise you will melt the plastic!

Not really it is possible to thread the wires all the way through the connector, then solder to the wires and push back into the plastic.
However, I found a video showing the way to do it so my question was answered.

If you insert the connector into the plastic and attempt to solder the wires, you will melt the plastic. You need to solder the wires to the connectors (away from the plastic) and then "snap" the connector into the plastic.

This punch tool makes the contact insertion into the housing simple:

https://www.progressiverc.com/ec3-assembly-punch.html

Thanks for the info. We ordered the EC3 and EC5 punches.

So did I.
Alan