grammar - Does [V-stem]に行く works for verbs whose stems are not stand-alone words by themselves?

I was taught that there is a grammatical form which goes like this: [V-stem]に行く

For example:　遊びに行く, which means "Go. Reason: to 遊ぶ"

I was wondering does this grammatical form work for verbs whose stems are not stand-alone words by themselves?

Ok, in case this gets confusing..

I mean 遊び is the stem of 遊ぶ, and 遊び is a standalone word in the dictionary which means "play (n)".

選び is the stem of 選ぶ, but　選び is not a standalone word in the dictionary. 選び by itself means nothing.

砕き is the stem of 砕く, 掘り of 掘る, 食べ of 食べる, 威張り of 威張る, 植え of 植える. but 砕き、掘り、食べ、威張り、植え by themselves are not standalone words in the dictionary. By themselves, they mean nothing.

Does these work:

1. 
   

   砕きにいく。 "Go. Reason: to 砕く"

   
   



2. 
   

   掘りにいく。 "Go. Reason: to 掘る"

   
   


3. 
   

   食べにいく。 "Go. Reason: to 食べる"

   
   


4. 
   

   威張りにいく。 "Go. Reason: to 威張る"

   
   


5. 
   
   

   植えにいく。 "Go. Reason: to 植える"

   
   

Answer

Yes, the form “[continuative form (-te stem) of a verb]に行く・来る” works even with a verb whose continuative form is not used as a noun in isolation. For example, 応援しに行く is perfectly fine even though the action of cheering is 応援, not 応援し.