Riding Shirts are no longer just tops. Many are now made from technical thermal fabrics that wick sweat and have anti-microbial material.
Base layers have become very popular in Australia. This is because they are long sleeved to protect us from the sun and cool enough to wear in summer. Brands with base layers include LeMieux, Ariat and Espoir.
LaNoir Saddleworld also has very well priced polos. The Huntington Phoebe for example has kwick dry technology. Wicking the moisture to the surface where it can quickly evaporate.
If you are wanting a Competition shirts see https://www.lanoirsaddleworld.com.au/product-category/competition-horse-riding-shirts/