Generally it is recommended to transmute units only if they are free units (who are generally designed to be transmuted) or 1-2 star units, some who can be obtained via normal summons but cost very little to transmute if you don't receive them by summoning.
However in some case it can be worth transmuting 5 star units.
3/Day Farmable Units[]
If the unit you wish to transmute is farmable with a 3/day shard quest, the best way to obtain them is to farm them until you get enough shards to transmute them and limit break them with the unit shards enough to put them to immediate use. You can check if the unit has a permanent 3/day farm on the Farmable Character Shards List - if you haven't yet unlocked their quest, you should do this as soon as possible so you can start farming them.
Although it may be tempting to transmute the unit as soon as you have 150 shards, it is generally not very useful to have a unit at only level 60, and it is quite possible you might pull the unit from a summon before you get enough shards to make them useful. Because of this, it is best to determine how many shards you need to put the unit into immediate use. For example, for Letitia or Judith, you would want 685 shards for transmute + max limit break + max gate 5 of enlightenment. Until you get that many shards you should keep farming the unit.
The transmute screen won't let you go to the quest after you have enough shards to transmute, so it is best to bookmark the unit under the story interface, or go to their quest in Babel Soul Shards, depending on where their quest is located.
1/Day or No Permanent Farm Units[]
5 Star limited units which are farmable 1/day or have no permanent farm generally require that you own the unit to transmute, so acquiring them via farming is not possible. They typically also do not appear in the Bazaar until you own them. Since most of these units do not have efficient banners, if possible you may wish to transmute them via purchasing unit shards, primarily via Louise in the Normal Shop.
Note: Memento obtained units, such as Voda, Tamis, Tehre, Fuu, Alma, Don Taras and Mocca can not be transmuted- you must obtain their limited memento to get the unit (and in the case of Tehre and Voda you also must limit break the memento the appropriate number of times).
Louise Shop[]
The Louise shop resets 5 times a day by default, and you can pay gems for resets- the price of resets gradually increases and resets daily. You can check the next reset time on the shop screen, or check the list of reset times and prices.
Each week the Louise Shop features different units- you can check the magnifying glass on the shop to determine what units are featured for the week. Typically there are 2 slots in the shop which can contain shards for limited units (each slot units will be listed at a different location under the magnifying glass), but they also typically also contain other normal pool units (usually 1-2 limited units out of 3-4 possible units in each slot).
Each time the shop resets, a random unit of the featured in that slot for the week will be available, at a random amount of shards (3, 5, 10, 15, 20 or 25). The shards are sold for 80 gems per shard, so if you need to obtain all 150 shards for the unit it would cost 12,000 gems total, plus any gems spent on refresh. However in many cases there are other sources available for shards to reduce the number you need to purchase.
Because the shop units change each week, and because new units often have a temporary 100 shard quest that you can only do if you own the unit, it may be worth paying for some refreshes (especially the cheaper ones) to ensure that you get enough shards before the unit you want leaves the shop. Typically most players only do the 100 gem reset twice daily until the last day, then pay for more refreshes as needed.
Not all limited units can appear in the Louise Shop. Genesis "god" units (ones designed to be strong without enlightenment such as Emmel, Nyx and Mira) do not appear there. Nero also does not appear there.
Bazaar[]
The Bazaar appears for 30 minutes after a certain amount of AP is spent and is on a 7 hour cooldown - the contents vary each time the cooldown expires. Similar to Louise, different units are featured each week. Although most high value limited units in the Bazaar require that you own them for their shards to appear, there are exceptions to this, primarily in the case of some of the Global Original units, which can appear even if you don't own them when featured. Typically the shard price is 80 gems per shard, the same as Louise Shop.
Other Shard Sources[]
If the unit you wish to purchase in Louise is new, they likely will also have other potential sources of shards. Generally new units are introduced with a new job trial mission, available for 2 weeks, which gives 25 shards for the unit. Sometimes they also come with a gem spent reward which will give shards of the unit based on how many gems you spend that week. There are also often bundles sold for money available with shards for brand new units.
Some units are associated with box events which may give another 25 shards, and possibly also have random drops of additional shards from the daily hard quests. Some events like collaborations may also give out random free shards for the units or bonus shards from banners. These may all help reduce your shard cost for transmuting the unit.
For units that do not appear in Louise, they sometimes have banners which give guaranteed 25 shards on a looping banner for the unit (usually 3 or 5 steps per loop). These are not cost effective banners, but may be a backup option to transmute the unit if you are willing to spend a lot of gems trying to summon the unit and fail to get lucky along the way.
Why Transmute instead of Summon?[]
Most limited unit banners do not have a good guarantee and have rather bad rates (0.5 to 0.66% featured rate) and cost 2,500 gems per step. So it can be extremely costly to acquire limited units from banners if you don't get lucky with your early pulls. Even if the banner gives 25 shards guaranteed on the 5th step for example, if you don't pull the unit that can run you 50-75k worth of gems depending on other shards available and your discount tickets.
If the unit is featured in the Louise Shop, you can have a more reliable means of acquiring the unit, potentially significantly cheaper than if you end up having to go a lot of steps to obtain them.
For example, a unit with a 25 shard new job trial and 25 shards in box event would only need 100 shards to transmute, which would bring the cost down to 8,000 gems plus any refresh needed. Even if you end up having to pay for refreshes (typically the first 2 refresh of 100 gems a day is cost efficient, then if you don't get enough shards, can pay more for refresh on the last day), it can end up being cheaper and more reliable than getting unlucky on a banner.