"becomes internalised" is correct and usable in written English.
You can use it when referring to something that has been deeply ingrained in the mind or spirit. For example, "The new values she learned in school became internalised over time."
The way grief becomes internalised in hunched shoulders and twisted limbs.
Thus, Netrins, which are transcribed within the VM, becomes internalised within the adjacent midgut cells in a Fra dependent manner.
HMFG1 becomes internalised after antigen binding (Aboud-Pirak et al, 1988) and thus provides a valuable tool for the selective delivery of cytotoxic agents into tumour cells.
As Kenan Malik, author of the forthcoming book From Fatwa to Jihad: The Rushdie Affair and Its Legacy, has written: "The fatwa has in effect become internalised".
Nanoparticles may also interact with and become internalised by dendritic cells, key antigen-presenting cells of the immune system, where a better understanding of these processes could pave the way for improved vaccination strategies.
Comments from students such as "makes you want to do your best" and "makes one strive to obtain high marks " seem to indicate that what might at first glance be perceived to be an external motivating factor may, for some, has become internalised, as they saw the bonus as an attainable goal.
Over the years the paintings became increasingly internalised and cryptic.
The dialectical tension that, in ancient tragedy, constituted the public life of the city, becomes privatised and internalised inside Hamlet's skull, leaving him divided and paralysed.
This internalised moment became Form (2013), a stack of blocks which still holds that feeling of introspection, but it has been expanded incrementally to the point where we reach a cube: Murmur (2014).
So I became an arsehole and internalised loads of shame about where I was raised, only to move to a privately-owned council flat in London and spend my evenings re-learning Welsh on Duolingo and getting bang into folklore about lakes.
New TMM structures of knowledge and a mutual frame of reference emerged during the intervention process and became more or less internalised by the participants in the study.
