Since it is OK to say "The house is becoming more expensive", become can be a dynamic verb for dynamic verbs allow the progressive tense. I wonder if become is also considered as a stative verb.
Become is a contraction of come to be, which is a general Inchoative (change of state) predicate,
like start, finish, end, begin, and continue.
- He became tired ~ He came to be tired - He was not tired at time t₀, and he was tired at (later) time t₁.
- He became a doctor ~ He came to be a doctor - not a doctor at time t₀, and doctor at (later) time t₁.
Inchoatives change states, but change between states may be gradual, allowing for optional continuous constructions, under the right circumstances. These all mean the same thing, for instance:
- The value of the house continues to rise every year.
- The value of the house is continuing to rise every year.
- The value of the house becomes higher every year.
- The value of the house is becoming higher every year.
I do not know what do you mean by dynamic but I think you mean action verb and a state verb
Become can be used in the progressive forms and other forms too.So it can be a state verb but may not be a dynamic verb though it is used in the progressive forms but know and understand can be pasivized though they are not usually used in the progressive forms.
Become can not be used in the passive voice.It is always intransitive like come and go and usually talks about a change of state.Anyhow, it is a special verb.
He became a doctor
He became famous
Previously, he was a doctor.Later he became a professor.(change of state in the past)
I will become a doctor.( future time)
English is becoming difficult for me day by day.( present progressive)
It is becoming cold( present progressive)
Unlike verbs concerning mind such as know, understand etc. become can be used in the progressive forms in all tenses.
Here is the link.