Learn Turkish: Göç edenler her zaman memleketlerini özlüyor. - Those who emigrate always miss their hometowns.

This uses the present participle suffix -en on the auxiliary verb etmek. Göç etmek means 'to emigrate'. Adding -en turns it into an adjective meaning 'who emigrates'. By adding the plural suffix -ler without a noun after it, it acts as a noun itself: 'the ones who emigrate' or 'those who emigrate'.

What are the suffixes on memleketlerini?

It breaks down like this: memleket (hometown) + -leri (their - third person plural possessive) + -n- (buffer letter) + -i (accusative case). The accusative case is needed here because it is the specific direct object of the verb özlemek (to miss).

Why is the verb özlüyor and not özleyor?

The dictionary form of the verb is özlemek (to miss). When you add the present continuous suffix -yor to a verb ending in e or a, that vowel narrows. Because of vowel harmony with the ö in the root, the e narrows to an ü, making it özlüyor.

The current lesson is about using zaman for 'when' clauses. Is her zaman acting as a 'when' clause here?

No. While you are learning to use zaman (time) to mean 'when' in this lesson (like in gideceğim zaman - 'when I will go'), here her zaman is simply the standard phrase for 'always' (literally 'every time'). It functions as a regular adverb of frequency, not a temporal clause.