What you are talking about is homophobia. Homophobia is only 
one aspect of conservativism.
If you were to say many minority communities are unfortunately homophobic, that would sadly be accurate. On that note, so are White Christians - statistically more so, in fact.
However, conservativism extends beyond homophobia. 
Thus why Hispanics, Muslims, etc. - despite homophobia being an issue - lean mostly towards being Democratic and liberal rather than conservative.
There are reports of some Hispanics being homophobic and of some white gay people being racist. The catch is though - those differences are put aside when it comes to the voting booth since they too are being targeted.
As shown, 60% of hispanics lean Democrat. Thus, there is no "trend conservative" as you attempted to say before.
Will more minorities vote Republican when that party stops focusing on White Supremacy (targeting every minority group in the country)? Absolutely. That day, however, sadly doesn't seem like it'll come any time soon - probably thirty to fifty years from now. If anything, Republicans are openly tripling down on it.
Intriguingly per the LGBTQIA+ front, Latino youth statistically is the mostly likely to identify as non-heterosexual. Thus, there is a trend there that is visibly changing from older generations.
	
	
		
			
				
			
			
				
				More than one in five Latino millennials identify as LGBTQ, according to a recent survey.
				
					
						
							
						
					
					www.nbcnews.com
				
 
			 
		 
	 
It should also be noted Latinos and Asian Christians are 
MORE likely to say homosexuality 
should be accepted than White Christians do.
	
	
		
			
				
			
			
				
				The Religious Landscape Study is a comprehensive survey of more than 35,000 Americans’ religious identities, beliefs and practices that’s been conducted in 2007, 2014 and 2023-24. Pew Research Center.
				
					
						
							
						
					
					www.pewresearch.org
				
 
			 
		 
	 
Statistically American Muslims are known to be 
less homophobic than White Evangelical Christians. As to not further false narratives out there that they are more so.
	
	
		
			
				
			
			
				
				While President Trump may claim he's protecting LGBT people from homophobia in the Muslim world, some of the biggest opponents of our rights are his fan base.
				
					
						
							
						
					
					www.advocate.com
				
 
			 
		 
	 
It's crucial to counter false narratives with actual statistics showing that those narratives aren't true or the norm.