Signed languages are natural languages, so in many cases their development in Asia is the same as their development in Europe. Where there is a large enough concentration of deaf people to develop a manual language, it happens naturally.

There are a few countries, though, where naturally developed sign languages are no longer in use. Modern Thai Sign Language is heavily influenced by American Sign Language, as educators who had been trained in the US established schools for deaf children in Thailand. ASL merged over time with existing sign languages in Thailand to create the Modern Thai Sign Language in use today. (This is actually how ASL was created, too - the natural amalgamation of French Sign Language with existing signed languages in the US.) Filipino Sign Language followed a similar path to Thai.

Taiwanese Sign Language and Korean Sign Language are both heavily influenced by Japanese Sign Language, from the period when Japan occupied these areas. There has been significant departure from JSL in both countries, but the influence is still visible.

Edit: Coming back with some other resources now that I'm not on mobile.

• There's a video explaining the history of Modern Thai Sign Language here, it's in ASL and Thai Sign Language but is currently uncaptioned. (If I can manage to caption the ASL portion at some point, I will do so via Amara.)
• For those with journal access, DOI 10.1353/sls.1996.0012 will pull up a 1996 article from the journal Sign Language Studies called "Modern Standard Thai Sign Language, Influence from ASL, and its Relationship to Original Thai Sign Varieties."
• The book Sign Languages in Contact has an article comparing the Taiwanese and Japanese sign lexicons.
• You can also read a bit more about the relationship between JSL, TSL, and KSL thanks to Google Books, which has this section of "Variation in East Asian sign language structures" from the Cambridge Language Survey of Sign Languages.