Like, were the knights wealthy at first but lost their wealth when they joined an order?
The term 'poor' (lat. pauper), associated with chivalric orders, although explicitly present only in the name of Poor Fellow-Soldiers of Christ and of the Temple of Solomon more commonly known as Templars was linked directly to the monastic vows taken by every monk upon joining the order, as stipulated by canon law. Upon getting official recognition on the Council of Troyes in 1129, Templars became an official militant monastic order with a rule similar to that of other orders that was later adopted by other similar organizations (Knights Hospitaller, Order of Lazarus, Teutonic Order etc.). It means that technically, despite its core being composed of nobles and distinctly military mission, Templars and later orders were organized much like any other monastic order, and its member had to observe the rules of chastity, obedience and poverty. Chastity meant basically celibacy (that was required of monks, but was not a norm among Catholic clergy as of early 12th century), obedience was an agreement to follow the heed of the superiors and poverty meant that the order members could either hold limited possessions or were forbidden from possessing any personal property (the details varied between the Orders and also fluctuated with time).
It is true that knight were generally members of the nobility and they had more or less wealth (with few exceptions they were at least local landowners), although stating that they were all rich and influential is quite the overstatement. However, vows of poverty meant that each candidate had to relinquish their worldly possessions. Of course, the Order preferred that the candidates ceded their wealth to the organization, but knights could have given part or even all of their wealth to the family before taking the vows (although most commonly they were donating at least part of what they had to the Order to bolster up the organization they were joining). According to the Temple Rules, later adopted by other Orders, all the possessions (from clothes to land) received from brothers and external donations were communal, i.e. did not belong to anyone in particular, but rather were considered the property of the Order and were managed by Order officials according to the needs of the brothers and other members of the Order. And although the Order itself could have been a large, affluent and influential organization, the knight-brothers, even though able to use the best equipment their times had to offer (and means to order such things with the approval of the appropriate officials), had very little to call their own, save for clothes (and event this could have been optional) and personal trinkets. Everything else, from horses to weapons and armours to money and servants were only assigned to them (not unlike like equipment in the modern army) and remained the property of the Order.