English Translation
The group of clowns began stumbling as they ran.
Picturing this scene in my mind, I see the clowns beginning to stumble one by one, so I had expected “empezaron a tropezar”. That would go well with the plural corrían, I believe. 
The subject of empezó is grupo, so the verb must be singular. Corrían is in a different clause so payasos can be interpreted as the subject there. The sentence is awkward and would work better with just ‘Los payasos’.
You do hear (rarely) things like ‘el grupo empezaron’ but I think these are better interpreted as ‘(en) el grupo empezaron’ or just by people changing their mind on what they want to say mid sentence.
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