For ～たる adjectives, the ～と ending is the adverbial -- same as the ～に ending for ～な adjectives.
The adverbial is often used right before a verb or another adjective, but it doesn't have to be. Sometimes an adverbial phrase is used to provide context or a shade of meaning for what comes after, even if that isn't a verb or another adjective.
Consider a similar adverbial construction in English:
Sheepishly, she lowered her hand and sat down.
The adverb "sheepishly" here doesn't really apply to any of the verbs in the following clause, and instead it serves as emotional coloration for the entire sentence.
In your sample Japanese text, the adverb 「淡々と」 can be parsed along similar lines, describing the mood or manner of the rest of the sentence, not just one of the specific verbs in that sentence.