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A person who converts loses his halachic connections to his family (see the gemara at the bottom of Yevamot 48b as well as Yevamot 98a).

What happens to "filial mitzvot" (e.g., filial respect/kibud av v'em, mourning/aveilut, being fruitful/pru u'rvu but also more broadly) when a parent and their child both convert?

Practically, imagine a person converting and a year later his father/mother also convert. Does that person "regain" an obligation to honor and mourn his newly Jewish parent? Do the Jewish children of the converted parent count as children from a pria v'rvia perspective?

Are there general rules or is it specific for each commandment?

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As far as the Mitzvah of Pru Uruvu, the Minchas Chinuch in Mitzvah Alef has a long piece on the topic

https://www.sefaria.org/Minchat_Chinukh.1.2?lang=bi

Here are 2 excerpts that show that he holds although the children are not considered "sons" they are considered his "offspring". With this differentiation, perhaps you could apply it to other mitzvos as well (i.e. if the mitzvah requires a child or just offspring)

נחזור לענינינו בנכרי ונתגייר ובניו עמו כיון דלא עשה בפטור רק ההכשר מצוה ועיקר המצוה נעשית בעת החיוב קיים מ"ע

ונראה דהא דתלינן ביחוס זרע אחריו היינו דוקא בעבד ועכו"ם דלכשנתגיירו ונשתחררו אין הבנים בנים שלהם לשום ענין מדיני התורה דגר שנתגייר כקטן שנולד דמי כמבואר בר"מ פט"ו לענין עריות וביבמות וסנהדרין ע"כ לענין פ"ו ג"כ לא נחשבו בנים רק בעכו"ם הגם דלא נחשבו בניו כיון דמתייחסים אחריו קיים המ"ע .ל

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