The simple answer would be: sketching.

The long answer would be that, however, the general interest for "capturing fleeting moments" was only developed as an artistic research after the invention of photography.

This is not to say that the idea of a "time" and "moment" was created in the 1800s. For instance, Italian Baroque art exhausted the idea of the "key moment" for dramatic purposes. In fact, if you look at Caravaggio's works, they all seem to focus on the very specific moment before a dramatic event occurs and it's usually all surrounding tragedy. Baroque artists also captured this idea through shadow manipulation, specially in northern-European still lives—but it's a general cultural aspect of Baroque painting and you can also see it in Las meninas by Velázquez, for instance. But even so, this was all manipulated. Artists wouldn't wait for dramatic events (murders, suicides, torture) to happen so they could sketch them. It was all their creativity, based on this principle of presenting (to the spectator) the precise moment before an activity results in tragedy.

"Capturing moments with fast action" was something that became an interest for artists at the same time and pace as people began to be interested in photography, somewhat as two sides of the same coin. It is the cultural paradigm of the French society in the 1800s that motivates this research. In the same path, you see that science is also interested in figuring out how eyes work and finally understanding that the eye is almost like our brain's camera lens.

The first painters to ever make the idea popular were the plein-air 1830s Barbizon school. Up until the 1800s it was not common for painters to paint outside of their studios, mainly because artists would have to work with their own pigments and it was not easily done outside as it involved mixing chemicals, so until the 1830s they would mostly sketch outside and do the rest of the work at home of combining, arranging, assembling, readjusting the colors, etc. In the 1830s, the rise in popularity of landscape plain-air painting was matched with premade paint in tubes and this simple thing allowed artists to go outside and paint without a hassle. However, artworks that were to be presented to the Academy had to follow certain rules of composition so everything collected during the artists' time outside remained for some time still a "sketch".

It was later on with the advances of modernism that this "sketch" became the "artwork" itself, specially with Impressionism. Impressionists are the ones who were after this idea of "capturing the moments" and they took full advantage of the technology they had. Most of these artists' works are small-sized because small sizes are easier to take out to the streets and paint directly onto (and also because most of them never made great fame, haha). This is the moment when the idea of "a view from place Y at time X" becomes popular in painting too.

Photography could not give such a response because it took a long time to develop and also photographic equipment was bulky and expensive. When it was first invented, it was used as a scientific tool and when it became popular, it was used for portraits rather than landscape or "moments". As cameras became lighter and more portable, towards the late 1800s, this idea changed, but at that time, painting had already achieved the idea of the "momentary". Painting won the race, I guess, haha.