BI+FT & immigration
The Basic Income would be a privilege of citizenship. Whilst providing support to refugees is a moral and legal duty (and to economic migrants may be a legal duty), that need not imply automatic entitlement to support in the same way as citizens would be automatically entitled to a Basic Income.
For those to whom it is not a birthright, citizenship should be earned. Those who have not yet earned citizenship should not be entitled to the Basic Income.
Entitlement gained through working
The test should be contribution to the economy of the UK (or whichever country was applying this system). Citizenship might, for instance, be granted whenever the individual has worked for at least two years within a consecutive three-year period (to avoid seasonal workers gradually accumulating rights to citizenship of a country to which they were not committed). That is a relatively short test for those who have arrived with realistic prospects of employment (once they have worked for two years, it would be clear that they would pass the test regardless of what happened in the third year, and citizenship could be granted). It would allow for occasional bouts of unemployment as can happen to anyone, without either setting too draconian a test or granting citizenship to immigrants who are not contributing to society.
Alternatively, citizenship might be granted when an immigrant had paid more than a threshold level of taxes within a defined period (say, £10,000 within 3 years). That would allow for those who can make a substantial contribution to the UK economy to gain citizenship rapidly, whilst enabling those on lower earnings still to gain citizenship if they contribute sufficient value over time. It would also deter gaming of the previous suggestion, where immigrants are paid a minimal amount officially, topped up with unofficial, untaxed, cash payments, to gain entitlement to citizenship whilst reducing the cost of employment below that of citizens.
Gainful employment measured by payment of taxes
The test of having worked would be payment of taxes (which would cover all legal workers, as there are no allowances and exemptions under a BI+FT system). All immigrants would have to register for a National Insurance number (or equivalent) on entry, work within the legal economy, and pay taxes on their earnings, if they wished to start accumulating the entitlement to citizenship of their adoptive country. This should substantially reduce the problem of immigrants working "on the black", undercutting native workers because the costs of illegal employment are lower than the costs of legal employment. If publicized externally, it should also help to address the problem of trafficking, because potential migrants would know that it would be essential for them to enter the country legally and be visible to the authorities, if they hoped to have a chance of gaining citizenship.
Correcting the balance in the job market
Entitlement to Basic Income would give citizens a competitive advantage in the job market, rather than the disadvantage they often experience at present. A citizen in receipt of a Basic Income would be in a position to accept a lower wage for a job than an immigrant without the benefit of a Basic Income (or live more comfortably on the same wage). If no citizen is prepared to do a job for the combination of Basic Income and wage that an immigrant is prepared to do for just the wage, it can be assumed that either there are no citizens with the appropriate skills seeking employment, or citizens with suitable skills who are not willing to work at that price have no grounds to complain against immigrants who will do the same job for less money, paying just as much tax and receiving no benefits.
Less bureaucracy and greater freedom of movement
Greater freedom of movement and less bureaucracy in the assessment of "suitable" immigrants should accompany this policy. The test of suitability would not be the ability to say the right things to bureaucrats, but whether you are coming with the genuine intent of working and therefore contributing. Some savings should be achieved from the reduction in the need for border police and controls.
Provision of conditional, targeted support for refugees
Some residual conditional benefits system may be required for refugees and (if necessary by law) economic migrants. It would depend whether charities could provide the necessary support if the numbers were limited to this smaller group of potential claimants.