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I'm rather confused by the question. It could mean:
Were there slaves in Asia during the US Civil Rights era (1964 on)?
Were there Asians who were slaves in the US during that era?
Were there Asians whose ancestors had been slaves in the US before the outlawing of slavery, and did the descendents participate in the freedom marches &c?
I'm even leaving out the sillier ones.
This is a very confusing question.
The 13th Amendment which abolished slavery was signed into law in 1865. Let's say slavery was effectively gone in the usual sense within a couple decades of that. The US Civil Rights Movement was around the 1950s and 1960s, nearly 100 years later. Not only were there not any Asian slaves in the US at that time, there weren't any slaves* in the US at that time.
Slavery was effectively done with long before the Civil Rights movement. The Civil Rights movement wasn't about slavery. It was about rights and equal treatment.
I'm not sure what you're getting at otherwise. It seems that you believe slavery was still going on in the 1960s, or that the Civil Rights movement was happening a century earlier than it did.
Could you possibly clear up what it is you're asking?
* Going by the usual definition, not getting into the issues of child labor and other things that could be called "slavery", since I think that's not what OP is getting at either.