Bring some good trail running shoes with you. Trail shoes differ from normal running shoes because of the other profile they have. Running on off-road tracks requires you to have a shoe with more grip and a stronger, more resistant sole. There are a lot of different types and brands and you should try some to know which one you like. Some well known brands for trail running shoes are: Salomon, La Sportiva, Scott, Inov-8, Merrell, Dynafit, Hoka, Brooks, Sauconyand Altra.
The weather in the Alps can change very quickly. Depending on the weather you can run with shorts and a t-shirt or top. In general always bring a wind or rain jacket with you. It can be colder the higher you get and especially on exposed areas like mountain passes or ridges the wind can be a lot stronger. Sometimes it is necessary to bring some gloves and a hat too or some thermal clothing.
A little running vest to put your food, water and extra clothing in. Personally I always put a rescue blanket in it as well. The same applies to running vests as to shoes. There are different brands and sizes and you better try them to see what fits you well. Well known brands are: Salomon, Inov-8, Ultimate Direction, Nathan and Raidlight.
Hiking poles if you prefer. Well known brands for hiking poles are: Leki and Black Diamond.
When you decide to sign up for a trail run I will send you a specific material list with the equipment you need to bring with you.