请客 is a verb-object-verbal phrase (联合词), therefore it is able to seperate both parts and put further structures in between. I read I could put 起 in between together with 来 at the end to form a so called "complement".
Now, all i know that i can build such a structure, for instance:
-分类 --> 分起类来
-吃惊 --> 吃起惊来
-贬值 --> 贬起值来
-游泳 --> 游起泳来
the problem is, i only know how to build this structure, but i can not find any explanation how the meaning is changed by it.
This infix -起-来 is usually considered a variation of the suffix -起来, and analysed as having an inceptive aspect, also called the inchoative aspect. The sense that it produces is "starting to do", "to begin to" etc.
起来 often interacts with 了 to form a particular viewpoint of an "accomplishment" action as having starting in the past, with the focus on the start point rather than as a whole.
It should be contrasted with 着, which does not emphasise the start of the action, but presents the action as one continuous state. Additionally, 起来 is different again from 下去, the latter of which is often called a continuative.
for meaning change due to 起来 following verb see any dictionary e。g。HSK dictionary （汉语水平考试词典）
In cases shown above verb is a 离合词 which just means that 起来 also is separated，but does not seem to change the generally valid effect of adding 起来 following verb。
起来（５）用在动词后面，表示动作完成，并有聚拢，合并在一起的意思 indicate gathering together or closing）：国家终于统一～了｜他把钱都藏～了
（６）用在动词后面，表示动作开始，并有继续下去的意思（indicate beginning and continuation of an action）：她开始唱～了｜才说几句就哭～了。
（７）用在部分动词后面，表示一种估计，评价的意思（indicate impression，view or estimation）：这个天气看～不会下雨了｜这道题目做～不会很难。
It seems（６） may apply to all ４ given examples
in my opinion, "請客" vs. "請起客來" is similar to "do something" vs. "start doing something".
I've seen this structure translated as "when" before. E.g:
他的右腿伤了，走起路来很疼 He hurt his right leg, and it's sore when he walks
