If I saw or heard a sentence like
“They are trying to push their responsibility out.”, I would be puzzled.
To “push” something “out” seems like it should mean
“promote” or “distribute”, as in
- Every few days,
Microsoft pushes out updates for Windows, Office, and other software.
- Last year,
Microsoft made an effort to push Windows 10 out to all Windows users.
This page — "Push out" a product at WordReference Forums —
ambiguously agrees with my interpretation.
The OP asks whether [to] push out [a product] means cancel or launch.
The most decisive respondent (panjandrum) says,
“… I understand "pushed out this product"
to mean that the product was brought to market sooner than originally planned,
or with unusual haste.”
Others opined that it could mean either, or something else altogether.
the Cambridge English Dictionary defines push something out as
to produce large quantities of something:
Companies are pushing out products at almost any price,
desperate to generate cash to survive.
This is similar to the paradigm
of Push technology and Pull technology in network communications.
According to Wikipedia,
Push technology, or server push
is a style of Internet-based communication
where the request for a given transaction
is initiated by the publisher or central server.
It is contrasted with pull/get,
where the request for the transmission of information
is initiated by the receiver or client.
Pull coding or client pull
is a style of network communication
where the initial request for data originates from the client,
and then is responded to by the server.
The reverse is known as push technology,
where the server pushes data to clients.
Also, “push [something] out” has a couple of biological applications,
which I won’t discuss except to suggest that the meaning might be analogous.
An idiom that approximates the meaning you want
(complete with the suggestion that
“they … give … their responsibility … to other people”)
with the voice (dynamic/activity level) of your candidate sentence is:
They are passing the buck.
(or “They are trying to pass the buck.”)
“Pass the buck” is defined as:
Cambridge English Dictionary:
to leave a difficult problem for someone else to deal with:
Don’t try to pass the buck – this is your responsibility, not mine.
Evade responsibility by passing it on to someone else.
to shift responsibility or blame to another person:
Never one to admit error, he passed the buck to his subordinates.
The Free Dictionary:
- to pass the blame (to someone else);
to give the responsibility (to someone else).
(See also The buck stops here.)
Don’t try to pass the buck!
It’s your fault, and everybody knows it.
Some people try to pass the buck whenever they can.
- to blame someone or to make them responsible
for a problem that you should deal with yourself