The three forms of "appear" give 3 different kind of voices in English:
"appearing" is for active voice (主動)
"appeared" is for passive voice (被動)
"to appear" is for action that is about to happen, but not yet.(將要)
With the above said, you shall choose "appearing", because these people arrive at the doorstep On THEIR OWN, not being brought to or already happening.
2013-02-08 11:08:54 補充：
"already happening" shall be reworded as "about to happen"!
2013-02-08 11:31:32 補充：
"not being brought to" or "about to happen" refers to "appeared" or "to appear".
2013-02-08 23:13:44 補充：
master Louis! Thank you for your approval!
2013-02-08 23:15:49 補充：
Certainly! You might ask why in your sentence, you use "appeared" is correct. The reason is "appeared" is NOT in a passive voice, but in a past tense (still active voice).
2013-02-08 23:18:06 補充：
The entire sentence is in past tense, therefore, when you rewrite "appearing on our doorstep", you shall also use past tense to make tense consistent. After all, it describes a situation within the same period of time.
2013-02-08 23:20:04 補充：
Your question is excellent! It shows that your using and understanding English is flexible, not restricted by grammar rules. As you progress in learning English, you will find that there are many ways to say the same thing. Have fun!