Someone being a political refugee from abroad gives them excellent qualifications to be welcomed into a new society, and equally good qualifications to have no influence over foreign policy.
A community of Cuban exiles who reside in the United States are essentially anti-communist hard line politicians who are not averse to demanding Americans be sent to war against a regime they dislike, as a result of their own family grievances. Many of those people fleeing Ukraine for Europe are likely to demand an aggressive line or even an open conflict with Russia by European countries.
If political or war refugees are allowed to hold political office, they will use it to make war speeches and basically pursue a vendetta at the expense of their adoptive country, and even their children are likely to also grow up to attempt the same.
Conflict of interest
One could argue that Cuban exiles and their descendants in the US, like Commonwealth residents present in the UK, are from territory formerly administered by the host country, and therefore we have some responsibility to listen to and act on their grievances. This argument, however, does not hold when those who were adopted are clearly putting their family interests over the interests of the adoptive country.
If someone is still loyal to the imperial power and re-joined its rule, e.g. if a Commonwealth resident is loyal to British rule, they should accept that the British know best and keep their heads down until we ask their opinion as part of an initiative launched by us, rather than claiming to know better and trying to pursue some form of authority over the British.
Let us consider another response. One could argue that indeed asylum seekers have taken their new nationality, and that it is therefore unfair of the country adopting them to point to them as different or less deserving of authority. However, the adoptees are the ones labelling themselves as different. If someone begins using their identity as a Cuban or a Jamaican to make their arguments regarding those lands, we can take it that they are giving up their new identity as American or British. They are forsaking our interests in favour of their own. They are therefore expressing conflicting loyalties, and if they have fewer rights as a result, they should have only themselves to blame.
Lobbying for revenge
It is important for a country to not be held hostage by another country or group from another country. It should be fine if the values and culture of the country change because of migration, and that some foreign sympathies arise because of it, but known individuals trying to manipulate our policy to support foreign interests should be distrusted. For this reason, foreign lobbies are inherently problematic because they call into question whether we are really allied to the other country or our representatives are merely being pressured by a hostile actor into supporting that other country.
Those who flee because they have no other homeland, and to whom we have historic or cultural obligations, should be welcome. If we have reason to think someone came to our country to advocate or lobby for a war or change to our policy regarding another country, though, then some thought should should be given to deporting them, and certainly they should not be given any kind of authority.
For those who want a moral liberal solution, rather than banning from office or deportation, a better course may simply be to examine these individuals who may advocate a foreign conflict. We should examine their loyalties with greater suspicion than we might examine others, in order to rule out a conflict of interest that causes them to secretly manipulate our country to achieve revenge against another country.