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How many challenges can be issued per assault?

In the BRB, in the introduction to challenges, it says "only one challenge can be issued per close combat." Further down the page, it says "Once one challenge has been made, no further challenges can be issued in that combat whilst the previous challenge is ongoing."

Which is it? If my Chaos Lord finishes up with one enemy character in a challenge, is it possible for him to move onto another in the same close combat?

It's both. "only one challenge can be issued per close combat." means that if you issue, and he refuses, you don't get to issue another one that turn to try and take out more characters. "Once one challenge has been made, no further challenges can be issued in that combat whilst the previous challenge is ongoing." means that if he accepts your challenge, no more challenges can be issued until the current one is resolved.

So yes, once you've finished off an enemy in a challenge you are free to issue another one next turn.

In fact, as a CSM player you must issue a challenge if you can - it's not an option.