How we change what others think, feel, believe and do
A person who uses passive-aggressive method to cope with stresses on them does this by 'attacking' others through passive means. Thus the aggressive intent is cloaked by the passive method.
Passive aggression often appears when a person is asked to do something which they want to avoid for some reason (such as priority of other work). By appearing to agree but not making any real commitment, they can avoid the action. A more severe form of passive aggression is to agree to commitments and then not do anything to fulfil them. A toned down version is to do the minimum possible whilst putting on a grand show of appearing to be fully engaged.
The 'aggressive' part of passive aggression is that while the person agrees to do something (they are often unable to say no) they resist in more subtle ways, creating problems while appearing to collaborate.
A person at a meeting is asked to complete a task with which they feel unable to comply. They talk at great length about it, discussing how important it is and all the various complexities that would be involved. At the end of the meeting, they still have not agreed to do anything.
A sales person uses a persuasive sales patter. The customer agrees that this is just what they want, but when it comes to signing the order, they find reasons why they cannot buy today.
A change manager asks people to change what they do. They agree but do not actually do what they agreed to do.
Passive aggression is a method often used by subordinates who are unable to directly oppose their superiors, and so need to resort to subtle and indirect means. It is also used with peers who can only ask (but not tell) them what to do, particularly where there is a false culture of supporting one team mates but the realities are that the day job takes a strong priority over helping one another.
This can also happen in a culture where it is impolite to say 'no' to a person's face. So people say yes, even when they mean no. 'Yes' in some cultures can mean 'I understand' but not 'I will comply with your request for action'.
Passive aggression may be rooted in childhood, where the impotent child cannot fight back against parents, teachers and other authority figures, and so resorts to truculence and withdrawal of commitment.
When someone keeps avoiding making commitments or appears to make a commitment to you but somehow does not comply, then you need to change the situation, otherwise you will not get anything done.
One way of handling this is to state very clearly what you want from them and then ask them directly (and repeatedly as necessary) whether they agree to do this (and by when).
Another approach is to 'name the game', pointing out to them what you are seeing in their behavior (do not accuse them -- just describe what you are seeing).