I'm a US citizen married to a Chinese citizen, and our child was born in the US and has a US passport. We know we need to apply for a Chinese Travel Document for our child to travel back to China. Our "local" embassy is Washington DC, but it's still about a 12 hour drive from where we live. We'll be spending time in New York this summer, and it would be so convenient to just get it done there. Are the embassies strict about only processing applications from within their jurisdiction? Do they require proof of residency?
According to Chinese embassy, Chinese citizens can apply for passport/travel documents at the nearest consulate, regardless of jurisdiction. (http://www.china-embassy.org/chn/lszj/zytz/t1572101.htm)
To be 100% sure, you could have your spouse contact New York Consulate or email Chinese embassy in Mandarin Chinese. They are pretty good about getting back to you.
I checked LA consulate website for travel document application requirement for your child's case. Although they also didn't mention anything about jurisdiction, proof of residency is required for the application. You will need application form, photos, child's birth certificate (original and copy), parents' passport (original and copy), parents' marriage certificate (orinial and copy), parent's proof of residency at time of child's birth (original and copy), and either you or your spouse should accompany the child to the application. (http://losangeles.china-consulate.org/chn/lszj/hzlxz/hzlxzxz/t1526294.htm)