Barclays (a brand we trust) put it this way:

Benefits of starting your own SIPP

• Choice – You won’t be restricted to pension funds offered by any single pension provider, but instead can invest in a broad range of investments from a range of different providers
• Tax-efficient – Your returns from investments within a SIPP are protected from income tax, tax on dividends, inheritance tax and capital gains tax (CGT)
• Flexible drawdown – You can pick from a wide range of options when you take your pension benefits, including a cash lump sum, a flexible or fixed level of income – or you can combine multiple options
• Tax-relief – You’ll receive tax relief at your marginal rate on an Annual Allowance, which for most people is £60,000 or 100% of your earnings, whichever is lower.1

A SIPP may be right for you if you’re confident making your own investment decisions and managing your pension payments against the relevant allowances.

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But how do you transfer to a new provider of a SIPP for you to manage yourself. Whether that is say AJ Bell, Hargreaves Lansdown or Barclays?

Again, let me quote Barclays: (By the way, I have my SIPP with HL.co.uk, and AJ Bell). I am quoting Barclays as they are a household name. Barclays say this:

Transfer pensions to us

If you have pensions elsewhere, you can transfer them to us at any time – but before you start, you’ll need a Barclays SIPP.

Transferring a pension doesn’t affect its tax-efficient status, but you should make sure that you're aware of all the risks and drawbacks involved. If you have access to a workplace pension, retaining that and transferring other existing pensions to it may be your best option, particularly if it offers the investment choices you’re looking for, as you’re likely to pay lower charges.

We will not accept transfers of defined benefit pensions (e.g. final or average salary pensions) or schemes that include safeguarded benefits (such as guaranteed annuity rates). If you’re unsure whether to transfer a pension, seek professional independent advice.

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Below I have included a factsheet from them. Simply put, you open a SIPP account with your chosen broker eg Barclays, and then fill in a transfer form and they do the rest.

This is what Hargreaves Lansdown say (HL.co.uk):

What to check before a pension transfer

• Ask your current provider if you'll need to pay any exit fees
• 
  
• Check you won't be forfeiting any valuable benefits or guarantees
• Make sure that all your personal details (including current name and address) are up to date with your existing provider
• Find out if your pension will be transferred as cash or stock (you might not have a choice)

You can transfer most types of pension to the HL SIPP:

• Personal and stakeholder pensions
• Pensions in drawdown
• Retirement Annuity Contracts (RACs)
• Self-Invested Personal Pensions
• Most Additional Voluntary Contribution plans (AVCs) including Free Standing AVCs
• Executive Pension Plans (EPPs)
• Most paid-up occupational money purchase pensions
• Old protected-rights pensions accrued from contracting out of the State Second Pension or SERPS (State Earnings Related Pension)

If you’re part of a defined-benefit (DB) pension, such as a ‘final salary’ scheme, transferring your pension to a personal plan is probably not in your best interest. These pensions not only give you a guaranteed income, they also normally offer benefits to a spouse or partner once you die. You might be able to transfer, but if the transfer value is more than £30,000, you’ll have to take advice from a regulated financial adviser and provide proof that the advice is in favour of transferring.

If your pension is transferred as cash, this means your provider will sell your pension investments, and transfer the cash amount. You will not be invested during the transfer, so you will not make losses or gains. You can buy investments once the transfer is complete. This usually takes 2-4 weeks, although some transfers can be complex and take longer. Timeframes also depend on actions from your existing provider.

If your pension is transferred as it is (invested in the stock market), your provider will transfer each investment and any cash. If you hold an investment that we don’t offer, we’ll contact you during the transfer to confirm your preference. This type of transfer can take several weeks, depending on your investments and provider. You stay invested during the transfer, so could make gains and losses. Usually, you cannot trade until the transfer completes. 

It may be possible to request a partial transfer of your pension to HL instead of being required to transfer the full amount. This can also allow you to transfer a portion of the pension from your current employment, while keeping the existing pension open to continue to receive contributions.

Not all providers can facilitate a partial transfer so we recommend that you check whether your current provider can before instructing a transfer with us.

What is Pension Wise?

Pension Wise is a free and impartial government service designed for everyone aged 50 or over, who has a UK-based personal or workplace pension. It aims to help you understand what type of pension you have, how you can access your savings and the potential tax implications of each option. https://www.moneyhelper.org.uk/en/pensions-and-retirement/pension-wise?source=pw