There isn't anything necessarily wrong with switching verb tenses in a sentence. It may often lead to awkwardness but it doesn't always.
In this case, there is no problem. In fact, there isn't really a switch in verb tenses at all.
The reason there isn't a problem in either of these sentences is because the second verbs used are both present participles. They end in -ing and take the same form regardless of any other verb tense:
A figure appeared at the door, waving happily.
A figure appears at the door, waving happily.
A figure will appear at the door, waving happily.
No matter what verb tense is used elsewhere, it will always be waving that is used.
The present participle is described as present, but it's actually present relative to whatever other verb is being used. For example, if a past tense verb is used, then the present participle is relative to it—and it takes place in the past.