Learn Danish: Vi er i gang med at gennemsøge den herinde, skuldertasken. - We are in the process of searching it thoroughly in here, the shoulder bag.

This is an example of right dislocation. The speaker uses the pronoun den during the main sentence, but tacks on skuldertasken at the end to clarify exactly what den refers to. It is very common in spoken Danish when you realize mid-sentence that the listener might not know what it is.

Since skuldertasken is delayed, how do we know to use den instead of det earlier in the sentence?

The pronoun must match the gender of the noun it stands in for, even if that noun comes later in the sentence. Because en taske (a bag) is a common gender noun, it takes den. If they were searching a house (et hus), the sentence would use det.

How does the prefix gennem- change the meaning of søge here?

While at søge means to search or look for something, adding the prefix gennem- (through) creates at gennemsøge. This implies searching through something completely or methodically, much like a police search of a bag or a building.

Why use er i gang med at instead of just saying vi gennemsøger?

Danish does not have an exact equivalent to the English continuous -ing form. While vi gennemsøger can mean both we search and we are searching, using the phrase at være i gang med at (to be in the process of) strongly emphasizes that the action is actively ongoing right at this very moment.