Learn Finnish: Etsittekö noita karkotettuja? - Are you looking for those banished ones?

This is a substantivized participle. In Finnish, you can take a participle like the passive past participle karkotettu (banished) and use it directly as a noun to mean "a banished person". Since it acts as a noun, it takes normal noun case endings. Here, it is pluralized and put into the partitive case.

What is the base verb for karkotettuja, and how did it get this specific form?

The base verb is karkottaa (to banish, exile, or deport). The passive past participle is karkotettu (banished). To make it plural partitive, the plural marker -i- is added, which turns the final -u into the partitive plural ending -uja, resulting in karkotettuja.

Why are noita and karkotettuja in the partitive case here?

The verb etsiä (to look for) strongly governs the partitive case. Because searching is inherently an ongoing, unbounded action (you haven't necessarily found the thing yet), its object must always be in the partitive. Both the pronoun and the substantivized participle must agree in case.

What is the base form of noita?

The base form is the demonstrative pronoun nuo (those). When declined into the partitive plural to match the case and number of karkotettuja, it becomes noita.