H1b - CAP subject to CAP Exempt to CAP subject again

Dear Group

An Indian consulting firm(CAP Subject) applied by H1b in 2012. I entered USA on that visa and changed private companies multiple times 2012 through 2016.

In 2016 I changed to a University employment. this cap exempt employer. It was a transfer of visa. they have extended that visa couple of times before my 6 year stay limit reached.

I continued my employment at same university on H4 EAD. While doing so, my employer filed for my PERM, I140 and re-applied for change of status from H4 to H1.

I now found a new employment with a private company and they are filing for my H1 transfer of employer.

Any such experiences in this group? Will this transfer possible without any hiccups? Anything I need to keep in mind and let my attorney know to facilitate this process run smoothly ?

Any inputs, greatly appreciated - Thank you!