Personally I would let them move. Especially if they have been there a long time. Finding a good tenant is costly and time consuming. If you have a good tenant who is willing to pay the higher rent, I think its a no brainer to try and keep them.
I just purchased a 12 unit apt building, all units are planned to be updated. The building was supposed to be delivered vacant yet seller did not do their part and I ended up with roughly 8 tenants. After talking to some about half are willing to move into renovated units and pay my market rent (which is significantly higher than what they were paying before). They liked the renovated units and most have been in the building for over 5 years.
Good tenants and they pay on time. Willing to pay what I am asking for just spending a little and renovating the units. This allows me to keep tenants (market rate), renovate their old units and get them on new leases. If I do not go this route, then trying to evict them will take additional time and money and may delay my renovation.