molecular biology - What is the mechanism of transgene integration (from expression vector to the host genome)?

What is your host in this case? For integration into the genome of a bacterium, you would need to use an "integration" vector. Most commercial vectors (such as pUC18) will be maintained without integrating into the host genome.

Here is a description of an integration vector from the Bacillus Genetic Stock Center to give you an idea of how foreign DNA would be integrated into a Bacillus genome:

Integration vectors are plasmids that feature conditional replication
coupled with a selectable marker. If the plasmid is transformed into
an appropriate host under conditions that select for the plasmid’s
presence but restrict its replication, all transformants will have
integrated the plasmid into their chromosome (or some other resident
DNA capable of replicating under the selective conditions). In
practice, the selectable marker usually specifies antibiotic
resistance. Conditional replication usually means that the plasmid has
replication functions that work in E. coli but not in gram-positive
bacteria, such as B. subtilis. Sometimes a temperature-sensitive
replication phenotype is employed instead. Integration is targeted to
a particular locus on the chromosome by including identical sequences
on the plasmid. If there is a single homologous sequence, a single
crossover will integrate the entire chromosome into the target locus
by a Campbell-type mechanism. If there are two homologous sequences,
and they are relatively close together on the chromosome, then a
double crossover will result in a cassette integrating between the
chromosomal targets. (source)

For most routine molecular biology labwork, there is no need to integrate genes into the E. coli genome. Integration is usually used to generate "knockout" cell lines to study gene function in the bacteria of interest.