Periphrastic verb forms, specifically a participle plus an auxiliary verb, are very common in English ("I am writing now"). They also appear in Latin and Ancient Greek and a number of Romance languages.
The types I'm most familiar with in Latin are the passives of the perfect system, and the gerundive of obligation.
Perfect passive participle + present = perfect passive
Perfect passive participle + imperfect = pluperfect passive
Perfect passive participle + future = future perfect passive
Future passive participle + present = future passive
This question indicates that you can also find perfect passive participle + perfect, to emphasize that an action is complete.
Are there any other attested combinations? For example, do we ever see a present participle combined with an auxiliary?