The process of evolution is ongoing in populations of living organisms. In a nutshell, the process of evolution involves the changes in populations of organism over time.
In the case of horse evolution, the process of evolution would have already been occurring in the organisms ancestral to modern horses, all the way up to and including modern horse populations. It's a continuous process.
Changes in organisms occur in DNA during reproduction (specifically in reproductive cells). In the case of sexual reproduction, organisms produce either sperm or eggs. During cell replication in the production of sperm and eggs, DNA is copied but the process is not perfect. Thus copying errors (called mutations) are introduced.
The resulting offspring is a combination of the DNA found in the sperm and eggs of the parents (offspring inherit approximately half of the DNA from each parent). Any mutations in the sperm and eggs will them make up the DNA of the offspring.
Depending on the mutations in the DNA, it may result in changes to the resulting offspring.
Of course, there are laboratory experiments on evolution all the time. As long as you have a population of organisms that reproduce, you have evolution.