Sadly, much of Ukraine is homophobic and does not believe in equal rights for members of the LGBT community. The human rights country report on Ukraine for 2013 noted country-wide reports of intolerance for gays. Many of those complaints included violence or the threat of violence based on a person's sexual identity. In addition, even Ukrainian law enforcement officials participated in harassment and failed to investigate hate crimes.
With Russia's de facto invasion, the situation has undoubtedly become worse. If LGBT activists hold pro-rights activities, Putin will use that as propaganda to sway Ukrainians away from the European Union. In Russia, LGBT persons are routinely assaulted. So, what choice would you make? Stay mute in the Ukraine until the Russian bear passes or come to the U.S. for a chance at a better life?
Yes, that's what they will do, too.
If you know anyone who is or may be seeking asylum, have that person contact the Broward law firm of Blandon Law at 954.385.0157.