Early Horses in the Czech Republic (Kyselý & Peške, 2016)

If Kyselý & Peške are correct, it's possible horses were being improved in Central Europe well before the early Funnel Beaker Culture and possibly as early as the wagon-wheeling Lengyel Culture.  Although larger outlier horses would have had to come from the Eastern sphere, these potential domestics would be as old or older than the oldest accepted domesticated horses of the Botai Culture.* 

They analyze a large set of post cranial bones beginning with the Magdalenian and look at successive periods to the Unetice then calculate the deltas in adult horses for various periods.  They conclude that within the Lengyel period there is the beginning of unnatural variance and a larger-than-to-be-expected population, suggesting human involvement.

(Commons)

One way to understand the variance argument is to look at the size range of domesticated dogs.  While there may be a place somewhere in the time-space continuum where English bulldogs could survive longer than a day on their own, it probably isn't a place that includes Tibetian mastiffs.  An archaeologist excavating a San Diego suburb thousands of years from now might also conclude that the Tibetian mastiff being excavated doesn't make sense for this ecology.  This is the problem with outlier Lengyel and TRB period horses and it is why human action is implicated.

Kyselý & Peške's dataset reveals that as the Eneolithic progresses through the Bronze Age the variability gradually decreases as well as the overall size of the horses.  This is evidence that the horse breed is being improved and neotenized, and this same trend can be seen in most domesticates.

I'll close with several factoids.  The Bell Beaker and Corded Ware horse remains are rather thin for this area, but they examine surrounding areas and conclude that the Hungarian Beaker horses were generally large (suggesting a horse with Eastern roots for this group), however with a great degree of variability, and possibly co-existing with more than one type of horse in the Carpathian Basin. 

- As the Late Bronze Age closes, horses are at their smallest, being called the 'Celtic Pony'.

- They don't believe environmental changes offer an appealing explanation for the changes in size or population, especially since those changes are not positive responses.

- There is little indication horses were used primarily for consumption and this is also not supported by age at death or sex.

Kyselý R. & Peške L. 2016. — Horse size and domestication: Early equid bones from the Czech Republic in the European context. Anthropozoologica 51 (1): 15-39. http://dx.doi.org/10.5252/az2016n1a2  [Link]

ABSTRACT
We collected and evaluated, by the ‘logarithmic size index’ (LSI) method, all available postcranial equid bones found in the Czech Republic from the Neolithic to the Early Bronze Age. Material from the Upper Paleolithic (Magdalenien) and Bohemian Late Bronze Age (Knovíz culture) was also included. Two different species of equids were documented: Equus hydruntinus Regalia, 1907 and Equus ferus Boddaert, 1785. The variation in the size of true horses was compared with data published for neighbouring countries. In most periods, the horses are found to be larger in the eastern part of
Central Europe than in the western part. The Czech lands appear to span the border of two worlds: the Pannonian plains and the western, geomorphologically diverse regions. The status of horses in the Neolithic Lengyel period from Moravia remains disputable. However, a high size variability in Eneolithic Funnel Beaker culture (TRB, 3800-3350 BC) together with a non-homogeneous distribution in Řivnáč culture (3100-2800 BC) and a significant increase in size between Lengyel and Baden-Řivnáč horizons (probably already in TRB) combined with the occasional occurrence of unexpectedly large individuals probably indicate the importation of tamed or even domesticated horses as early as the times of TRB culture, which is earlier than claimed in other recent studies, and possibly reflect multiple origins of the horse population. The large variability and repeated diminution in size of horses in the Early Bronze Age (Únětice culture, 2200-1700 BC) could indicate advanced domestication or multiple origins of the populations (or both). The persistence of wild horses in the Early Bronze Age cannot be proved osteometrically, but the presence of domesticated horses is considered certain.

See also Horse of Botai [here]