The LA homeless survey does track where people lived when they became homeless. In 2019, the number who became homeless outside of CA is 20% [1]. AFAIK, they don't release raw data so we can segment by newly homeless, but in 2018, this same figure was 14% [2].

Seattle / King County's survey asks a similar question and it's just 5% [3]. Conservative media up here likes to claim that homeless immigration is a significant problem (it isn't), but it seems like LA is facing a different challenge.

1 - https://www.lahsa.org/documents?id=3437-2019-greater-los-ang...

2 - https://www.lahsa.org/documents?id=2059-2018-greater-los-ang...

3 - http://allhomekc.org/wp-content/uploads/2019/07/Updated-7.11...